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1.
Let C be a closed convex subset of a uniformly smooth Banach space E and let T:C→C be a nonexpansive mapping with a nonempty fixed points set. Given a point u∈C, the initial guess x0∈C is chosen arbitrarily and given sequences , and in (0,1), the following conditions are satisfied:
- (i)
- ;
- (ii)
- αn→0, βn→0 and 0<a?γn, for some a∈(0,1);
- (iii)
- , and . Let be a composite iteration process defined by
2.
Christopher Mouron 《Topology and its Applications》2009,156(3):558-576
Suppose that is a collection of disjoint subcontinua of continuum X such that limi→∞dH(Yi,X)=0 where dH is the Hausdorff metric. Then the following are true:
- (1)
- X is non-Suslinean.
- (2)
- If each Yi is chainable and X is finitely cyclic, then X is indecomposable or the union of 2 indecomposable subcontinua.
- (3)
- If X is G-like, then X is indecomposable.
- (4)
- If all lie in the same ray and X is finitely cyclic, then X is indecomposable.
3.
Tingxiu Wang 《Journal of Mathematical Analysis and Applications》2006,324(2):982-991
With the Lyapunov second method, we study the abstract functional differential equation, . We obtain inequalities of solutions and exponential stability with conditions like:
- (i)
- ,
- (ii)
- .
4.
Zhi-Wei Sun 《Journal of Number Theory》2007,124(1):57-61
By some extremely simple arguments, we point out the following:
- (i)
- If n is the least positive kth power non-residue modulo a positive integer m, then the greatest number of consecutive kth power residues mod m is smaller than m/n.
- (ii)
- Let OK be the ring of algebraic integers in a quadratic field with d∈{−1,−2,−3,−7,−11}. Then, for any irreducible π∈OK and positive integer k not relatively prime to , there exists a kth power non-residue ω∈OK modulo π such that .
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7.
Prem L. Sharma 《Discrete Mathematics》2008,308(24):6003-6008
Given a cubical box C2n+1 of side 2n+1 and a supply of 1×2×4 bricks, it is proved that if n≥2, then
- (A1)
- one can pack bricks for n odd, and bricks for n even,
- (A2)
- the capacity of C2n+1 is , and if n≡1 or 2 (mod4), this upper bound for the capacity can be reduced by 1.
8.
Dejan Brcanov 《Discrete Mathematics》2010,310(19):2550-2554
Let and be two n-tuples of nonnegative integers. An all-4-kings n-partite tournament T(V1,V2,…Vn) is said to have a -property if there exists an n-partite tournament T1(W1,W2,…,Wn) such that for each i∈{1,…,n}:
- (1)
- Vi⊆Wi;
- (2)
- exactly ti 4-kings of Vi are not 4-kings in T1;
- (3)
- exactly ci 4-kings of Wi are not vertices of Vi.
9.
Edith Hemaspaandra Lane A. Hemaspaandra Stanis?aw P. Radziszowski 《Discrete Applied Mathematics》2007,155(2):103-118
We investigate the relative complexity of the graph isomorphism problem (GI) and problems related to the reconstruction of a graph from its vertex-deleted or edge-deleted subgraphs (in particular, deck checking (DC) and legitimate deck (LD) problems). We show that these problems are closely related for all amounts c?1 of deletion:
- (1)
- , , , and .
- (2)
- For all k?2, and .
- (3)
- For all k?2, .
- (4)
- .
- (5)
- For all k?2, .
10.
Yankui Song 《Topology and its Applications》2012,159(5):1462-1466
11.
We present a new construction of non-classical unitals from a classical unital U in . The resulting non-classical unitals are B-M unitals. The idea is to find a non-standard model Π of with the following three properties:
- (i)
- points of Π are those of ;
- (ii)
- lines of Π are certain lines and conics of ;
- (iii)
- the points in U form a non-classical B-M unital in Π.
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14.
Su Gao 《Advances in Mathematics》2008,217(2):814-832
A group G?Sym(N) is cofinitary if g has finitely many fixed points for every g∈G except the identity element. In this paper, we discuss the definability of maximal cofinitary groups and some related structures. More precisely, we show the following two results:
- (1)
- Assuming V=L, there is a set of permutations on N which generates a maximal cofinitary group.
- (2)
- Assuming V=L, there is a mad permutation family in Sym(N).
15.
Zongming Guo 《Journal of Differential Equations》2007,240(2):279-323
We consider the following Cauchy problem with a singular nonlinearity
- (P)
-
16.
Let H be a complex separable Hilbert space and L(H) denote the collection of bounded linear operators on H. An operator A in L(H) is said to be a Cowen-Douglas operator if there exist Ω, a connected open subset of complex plane C, and n, a positive integer, such that
- (a)
- (b)
- for z in Ω;
- (c)
- ; and
- (d)
- for z in Ω.
17.
Wen-Chung Huang 《Discrete Mathematics》2006,306(13):1351-1357
Let {n;b2,b1} denote the class of extended directed triple systems of the order n in which the number of blocks of the form [a,b,a] is b2 and the number of blocks of the form [b,a,a] or [a,a,b] is b1. In this paper, we have shown that the necessary and sufficient condition for the existence of the class {n;b2,b1} is b1≠1, 0?b2+b1?n and
- (1)
- for ;
- (2)
- for .
18.
Chuan-Fu Yang 《Journal of Mathematical Analysis and Applications》2010,365(2):742-554
In this work, we consider inverse spectral problems for the Sturm-Liouville differential operator on a d-star-type graph with standard matching conditions in the internal vertex, where the integer d?2. By using the Yurko's method (Yurko (2008) [27], Yurko (2009) [28]) we show that
- (1)
- if the potential function qj(x) on a fixed edge ej is prescribed on the interval , then the reciprocal of d of the spectrum suffices to determine qj(x) on the whole interval [0,π];
- (2)
- the 2 over d of the spectrum suffices to determine qj(x) on a fixed edge ej.
19.
Linus Kramer 《Advances in Mathematics》2005,193(1):142-173
Let G be a connected semisimple Lie group with at least one absolutely simple factor S such that and let Γ be a uniform lattice in G.
- (a)
- If CH holds, then Γ has a unique asymptotic cone up to homeomorphism.
- (b)
- If CH fails, then Γ has 22ω asymptotic cones up to homeomorphism.
20.