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1.
F. A. Dudkin 《Siberian Mathematical Journal》2014,55(1):72-77
A finitely generated group G that acts on a tree so that all vertex and edge stabilizers are infinite cyclic groups is called a generalized Baumslag-Solitar group or GBS-group. Let p and q be coprime integers other than 0, 1, and ?1. We prove that the Baumslag-Solitar group BS(p, q) embeds into G if and only if the equation x ?1 y p x = y q is solvable in G for y ≠ 1; i.e., $\tfrac{p} {q} $ ∈ Δ(G), where Δ is the modular homomorphism. 相似文献
2.
Reese Scott 《Journal of Number Theory》2004,105(2):212-234
Using a theorem on linear forms in logarithms, we show that the equation px−2y=pu−2v has no solutions (p,x,y,u,v) with x≠u, where p is a positive prime and x,y,u, and v are positive integers, except for four specific cases, or unless p is a Wieferich prime greater than 1015. More generally, we obtain a similar result for px−qy=pu−qv>0 where q is a positive prime, . We solve a question of Edgar showing there is at most one solution (x,y) to px−qy=2h for positive primes p and q and positive integer h. Finally, we use elementary methods to show that, with a few explicitly listed exceptions, there are at most two solutions (x,y) to |px±qy|=c and at most two solutions (x,y,z) to px±qy±2z=0, for given positive primes p and q and integer c. 相似文献
3.
Mervan Paši? 《Journal of Mathematical Analysis and Applications》2011,381(1):27-2293
Asymptotic and oscillatory behaviours near x=0 of all solutions y=y(x) of self-adjoint linear differential equation (Ppq): ′(py′)+qy=0 on (0,T], will be studied, where p=p(x) and q=q(x) satisfy the so-called Hartman-Wintner type condition. We show that the oscillatory behaviour near x=0 of (Ppq) is characterised by the nonintegrability of on (0,T). Moreover, under this condition, we show that the rectifiable (resp. unrectifiable) oscillations near x=0 of (Ppq) are characterised by the integrability (resp. nonintegrability) of on (0,T). Next, some invariant properties of rectifiable oscillations in respect to the Liouville transformation are proved. Also, Sturm?s comparison type theorem for the rectifiable oscillations is stated. Furthermore, previous results are used to establish such kind of oscillations for damped linear second-order differential equation y″+g(x)y′+f(x)y=0, and especially, the Bessel type damped linear differential equations are considered. Finally, some open questions are posed for the further study on this subject. 相似文献
4.
Preda Mih?ilescu 《Journal of Number Theory》2006,118(1):123-144
Catalan's conjecture states that the equation xp−yq=1 has no other integer solutions but 32−23=1. We investigate the consequences of existence of further solutions (with odd prime exponents p,q) upon the relative class group of the pth cyclotomic extension. We thus obtain several new results which merge into the condition
5.
Ali Reza Ashrafi 《Journal of Applied Mathematics and Computing》2002,10(1-2):167-174
A groupG is said to be (l, m, n)-generated if it is a quotient group of the triangle groupT(p,q,r)=<x,y,z?x p =y q =z r =xyz=1>. In [15], the question of finding all triples (l, m, n) such that non-abelian finite simple groups are (l, m, n)-generated was posed. In this paper we partially answer this question for the sporadic groupHe. We continue the study of (p, q, r)-generations of the sporadic simple groups, where,p, q, r are distinct primes. The problem is resolved for the Held groupHe. 相似文献
6.
Boris Širola 《Acta Mathematica Hungarica》2004,104(1-2):127-142
Consider these two types of positive square-free integers d≠ 1 for which the class number h of the quadratic field Q(√d) is odd: (1) d is prime∈ 1(mod 8), or d=2q where q is prime ≡ 3 (mod 4), or d=qr where q and r are primes such that q≡ 3 (mod 8) and r≡ 7 (mod 8); (2) d is prime ≡ 1 (mod 8), or d=qr where q and r are primes such that q≡r≡ 3 or 7 (mod 8). For d of type (2) (resp. (1)), let Π be the set of all primes (resp. odd primes) p∈N satisfying (d/p) = 1. Also, let δ :=0 (resp. δ :=1) if d≡ 2,3 (mod 4) (resp. d≡ 1 (mod 4)). Then the following are equivalent: (a) h=1; (b) For every p∈П at least one of the two Pellian equations Z 2-dY 2 = ±4δ p is solvable in integers. (c) For every p∈П the Pellian equation W 2-dV 2 = 4δ p 2 has a solution (w,v) in integers such that gcd (w,v) divides 2δ. 相似文献
7.
Svatoslav Staněk 《Journal of Differential Equations》1978,30(3):287-295
This paper gives a generalization of the Sturm comparison theorem for differential equations (p): y″ = p(t)y, (q): y″ = q(t)y under the assumption that the function p ? q changes its sign exactly once on [a, b] or ∝tbp ? q, ∝atp ? q maintain the sign on [a, b]. The results are used for investigating the distributions of zeros of solutions and the derivative of solutions of (p), (q). 相似文献
8.
The main result of this paper is that point sets of PG(n, q 3), q = p h , p ≥ 7 prime, of size less than 3(q 3(n?k) + 1)/2 intersecting each k-space in 1 modulo q points (these are always small minimal blocking sets with respect to k-spaces) are linear blocking sets. As a consequence, we get that minimal blocking sets of PG(n, p 3), p ≥ 7 prime, of size less than 3(p 3(n?k) + 1)/2 with respect to k-spaces are linear. We also give a classification of small linear blocking sets of PG(n, q 3) which meet every (n ? 2)-space in 1 modulo q points. 相似文献
9.
If an isometric embeddingl p m →l q n with finitep, q>1 exists, thenp=2 andq is an even integer. Under these conditions such an embedding exists if and only ifn?N(m, q) where $$\left( {\begin{array}{*{20}c} {m + q/2 - 1} \\ {m - 1} \\ \end{array} } \right) \leqslant N(m,q) \leqslant \left( {\begin{array}{*{20}c} {m + q - 1} \\ {m - 1} \\ \end{array} } \right).$$ To construct some concrete embeddings, one can use orbits of orthogonal representations of finite groups. This yields:N(2,q)=q/2+1 (by regular (q+2)-gon),N(3, 4)=6 (by icosahedron),N(3, 6)?11 (by octahedron), etc. Another approach is based on relations between embeddings, Euclidean or spherical designs and cubature formulas. This allows us to sharpen the above lower bound forN(m, q) and obtain a series of concrete values, e.g.N(3, 8)=16 andN(7, 4)=28. In the cases (m, n, q)=(3, 6, 10) and (3, 8, 15) some ε-embeddings with ε ~ 0.03 are constructed by the orbit method. The rigidness of spherical designs in Bannai's sense and a similar property for the embeddings are considered, and a conjecture of [7] is proved for any fixed (m, n, q). 相似文献
10.
Let E = Eσ : y2 = x(x + σp)(x + σq) be elliptic curves, where σ = ±1, p and q are primenumbers with p+2 = q. (i) Selmer groups S(2)(E/Q), S(φ)(E/Q), and S(φ)(E/Q) are explicitly determined,e.g. S(2)(E+1/Q)= (Z/2Z)2, (Z/2Z)3, and (Z/2Z)4 when p ≡ 5, 1 (or 3), and 7(mod 8), respectively. (ii)When p ≡ 5 (3, 5 for σ = -1) (mod 8), it is proved that the Mordell-Weil group E(Q) ≌ Z/2Z Z/2Z,symbol, the torsion subgroup E(K)tors for any number field K, etc. are also obtained. 相似文献
11.
This paper begins with a short historical survey on Catalan's equation, namely xp-yq=1, where p andq are prime numbers and x, y are non-zero rational integers. It is conjectured that the only solution is the trivial solution 32-23=1. We prove that there is no non-trivial solution with p orq smaller than 30000. The tools to reach such a result are presented. A crucial role is played by a recent estimate of linear forms in two logarithms obtained by Laurent, Mignotte and Nestrenko. The criteria used are also quite recent. We give information on the enormous amount of computation needed for the verification. 相似文献
12.
Xuefeng WangAihua W. Wood 《Journal of Mathematical Analysis and Applications》2002,267(1):361-368
We show that entire positive solutions exist for the semilinear elliptic system Δu = p(x)vα, Δv = q(x)uβ on RN, N ≥ 3, for positive α and β, provided that the nonnegative functions p and q are continuous and satisfy appropriate decay conditions at infinity. We also show that entire solutions fail to exist if the functions p and q are of slow decay. 相似文献
13.
《Journal of Number Theory》1986,23(2):219-237
It is known that a certain class of [n, k] codes over GF(q) is related to the diophantine equation y2 = 4qn + 4q + 1 (1). In Parts I and II of this paper, two different, and in a certain sense complementary, methods of approach to (1) are discussed and some results concerning (1) are given as applications. A typical result is that the only solutions to (1) are (y, n) = (5, 1), (7, 2), (11, 3) when q = 3 and (y, n) = (2q + 1, 2) when q = 3f, f >- 2. 相似文献
14.
《Journal of Algebra》2002,247(1):244-267
J. Chuang, R. Kessar, and J. Rickard have proved Broué's Abelian defect group conjecture for many symmetric groups. We adapt the ideas of Kessar and Chuang towards finite general linear groups (represented over non-describing characteristic). We then describe Morita equivalences between certain p-blocks of GLn(q) with defect group Cpα × Cpα, as q varies (see Theorem 2). Here p and q are coprime. This generalizes work of S. Koshitani and M. Hyoue, who proved the same result for principal blocks of GLn(q) when p = 3, α = 1, in a different way. 相似文献
15.
In this paper, we study global positive C4 solutions of the geometrically interesting equation: Δ2u+u−q=0 with q>0 in R3. We will establish several existence and non-existence theorems, including the classification result for q=7 with exactly linear growth condition. 相似文献
16.
S. D. Fisher 《Constructive Approximation》1996,12(4):463-480
Let Ω be a finitely-connected planar domain and μ be a positive measure with compact supportE in Ω. LetA p be the unit ball of the Hardy spaceH p. The main result of this paper is that Kolmogorov, Gelfand, and linearn-widths ofA p inL q are comparable in size to each other and to the sampling error ifq≤p. Moreover, ifp=q=2 andE is small enough, then all these quantities are equal. 相似文献
17.
M. Sh. Shabozov 《Mathematical Notes》1996,59(1):104-111
We find the exact value of the expression $$\varepsilon ^{(l,q)} {\mathbf{ }}(W^{(r,s)} ){\mathbf{ }}H^{w_1 ,w_2 } (G)) = \sup \{ ||f^{(l,q)} ( \cdot {\mathbf{ }}, \cdot ) - S_{1,1}^{(l,q)} (f;{\mathbf{ }} \cdot {\mathbf{ }}, \cdot )||_{C(G)} :f \in W^{(r,{\mathbf{ }}s)} H^{w_1 ,w_2 } (G)\} ,$$ , where? (l,q) (x,y)=? 1+q ?/?x l ?y q (l, q=0, 1, 1≤l+q≤2) andS 1,1(f; x, y) is a bilinear spline interpolatingf(x, y) in the nodes of the grid Δ mn =Δ m x ×Δ n y with Δ m x :x i =i/m (i=0, ..., m) and Δ n y :y j =j/n (j=0, ..., n). Here $(W^{(r,s)} ){\mathbf{ }}H^{w_1 ,w_2 } (G)$ is the class of functionsf(x, y) with continuous derivativesf (r,s)(x, y) (r, s=0, 1, 1≤r+s≤2) on the squareG=[0, 1]×[0, 1] and with the modulus of continuity satisfying the inequalityω(f (r,s);t, τ)≤ω 1 (t)+ω 2 (τ), whereω 1 (τ) andω 2 (τ) are the given moduli of continuity. 相似文献
18.
In [7], for the casesq even andq=3, a characterisation of the Buekenhout-Metz unitals inPG(2,q
2) was given. We complete this characterisation by proving the result forq>3. 相似文献
19.
Teodor Banica 《Journal of Functional Analysis》2011,260(11):3252-3282
We introduce and study natural two-parameter families of quantum groups motivated on one hand by the liberations of classical orthogonal groups and on the other by quantum isometry groups of the duals of the free groups. Specifically, for each pair (p,q) of non-negative integers we define and investigate quantum groups O+(p,q), B+(p,q), S+(p,q) and H+(p,q) corresponding to, respectively, orthogonal groups, bistochastic groups, symmetric groups and hyperoctahedral groups. In the first three cases the new quantum groups turn out to be related to the (dual free products of ) free quantum groups studied earlier. For H+(p,q) the situation is different and we show that , where the latter can be viewed as a liberation of the classical isometry group of the p-dimensional torus. 相似文献
20.
In this paper we establish existence of solutions of singular boundary value problem ?(p(x)y ′(x))′=q(x)f(x,y,py′) for 0<x≤b and $\lim_{x\rightarrow0^{+}}p(x)y^{\prime}(x)=0$ , α 1 y(b)+β 1 p(b)y ′(b)=γ 1 with p(0)=0 and q(x) is allowed to have integrable discontinuity at x=0. So the problem may be doubly singular. Here we consider $\lim_{x\rightarrow0^{+}}\frac{q(x)}{p'(x)}\neq0$ therefore $\lim_{x\rightarrow0^{+}}p(x)y'(x)=0$ does not imply y′(0)=0 unless $\lim_{x\rightarrow0^{+}}f(x,y(x),p(x)y'(x))=0$ . 相似文献