共查询到20条相似文献,搜索用时 15 毫秒
1.
Ryuichi Tanaka 《Topology and its Applications》2009,156(5):932-938
A CW complex B is described as I-trivial if there does not exist a Z2-map from Si−1 to S(α) for any vector bundle α over B and any integer i with i>dimα. For n>1, we determine all positive integers m for which the stunted projective space is I-trivial, where F=R,C or H. 相似文献
2.
Ryuichi Tanaka 《Topology and its Applications》2007,154(8):1827-1833
For a vector bundle α, let indα denote the largest integer m for which there exists a Z/2-map from Sm−1 to S(α). We prove that the equality indα=dimα holds for every vector bundle α over the complex Sn−1k∪en, where n?2 and k≠0, if and only if either k is even and n≠2,3,4,8 or k is odd. 相似文献
3.
Ryuichi Tanaka 《Topology and its Applications》2008,155(15):1687-1693
It is a classical theorem of Milnor that for every vector bundle over Sn, all the Stiefel-Whitney classes vanish if and only if n≠1,2,4,8. We describe a space B as W-trivial (except for one dimension) if for every vector bundle over B, all the Stiefel-Whitney classes vanish (except for a single fixed dimension). We establish theorems which state that certain high-connectivities of B imply these trivialities as well as a theorem which states that there are infinitely many “W-trivial except for one dimension” spaces. 相似文献
4.
Evelyn L. Hart 《Topology and its Applications》2006,153(9):1399-1420
Let denote a periodic self map of minimal period m on the orientable surface of genus g with g>1. We study the calculation of the Nielsen periodic numbers NPn(f) and NΦn(f). Unlike the general situation of arbitrary maps on such surfaces, strong geometric results of Jiang and Guo allow for straightforward calculations when n≠m. However, determining NPm(f) involves some surprises. Because fm=idFg, fm has one Nielsen class Em. This class is essential because L(idFg)=χ(Fg)=2−2g≠0. If there exists k<m with L(fk)≠0 then Em reduces to the essential fixed points of fk. There are maps g (we call them minLef maps) for which L(gk)=0 for all k<m. We show that the period of any minLef map must always divide 2g−2. We prove that for such maps Em reduces algebraically iff it reduces geometrically. This result eliminates one of the most difficult problems in calculating the Nielsen periodic point numbers and gives a complete trichotomy (non-minLef, reducible minLef, and irreducible minLef) for periodic maps on Fg.We prove that reducible minLef maps must have even period. For each of the three types of periodic maps we exhibit an example f and calculate both NPn(f) and NΦn(f) for all n. The example of an irreducible minLef map is on F4 and is of maximal period 6. The example of a non-minLef map is on F2 and has maximal period 12 on F2. It is defined geometrically by Wang, and we provide the induced homomorphism and analyze it. The example of an irreducible minLef map is a map of period 6 on F4 defined by Yang. No algebraic analysis is necessary to prove that this last example is an irreducible minLef map. We explore the algebra involved because it is intriguing in its own right. The examples of reducible minLef maps are simple inversions, which can be applied to any Fg. Using these examples we disprove the conjecture from the conclusion of our previous paper. 相似文献
5.
Duong Quoc Viet 《Journal of Pure and Applied Algebra》2009,213(11):2104-2116
Let A be a Noetherian local ring with the maximal ideal m and an m-primary ideal J. Let S=?n≥0Sn be a finitely generated standard graded algebra over A. Set S+=?n>0Sn. Denote by FJ(S)=?n≥0→(Sn/JSn) the fiber cone of S with respect to J. The paper characterizes the multiplicity and the Cohen-Macaulayness of FJ(S) in terms of minimal reductions of S+. 相似文献
6.
Michael Cole 《Topology and its Applications》2006,153(7):1016-1032
We prove that if a category has two Quillen closed model structures (W1,F1,C1) and (W2,F2,C2) that satisfy the inclusions W1⊆W2 and F1⊆F2, then there exists a “mixed model structure” (Wm,Fm,Cm) for which Wm=W2 and Fm=F1. This shows that there is a model structure for topological spaces (and other topological categories) for which Wm is the class of weak equivalences and Fm is the class of Hurewicz fibrations. The cofibrant spaces in this model structure are the spaces that have CW homotopy type. 相似文献
7.
Ioannis Souldatos 《Annals of Pure and Applied Logic》2012,163(3):225-237
We say that a countable model M completely characterizes an infinite cardinal κ, if the Scott sentence of M has a model in cardinality κ, but no models in cardinality κ+. If a structure M completely characterizes κ, κ is called characterizable. In this paper, we concern ourselves with cardinals that are characterizable by linearly ordered structures (cf. Definition 2.1).Under the assumption of GCH, Malitz completely resolved the problem by showing that κ is characterizable if and only if κ=ℵα, for some α<ω1 (cf. Malitz (1968) [7] and Baumgartner (1974) [1]). Our results concern the case where GCH fails.From Hjorth (2002) [3], we can deduce that if κ is characterizable, then κ+ is characterizable by a densely ordered structure (see Theorem 2.4 and Corollary 2.5).We show that if κ is homogeneously characterizable (cf. Definition 2.2), then κ is characterizable by a densely ordered structure, while the converse fails (Theorem 2.3).The main theorems are (1) If κ>2λ is a characterizable cardinal, λ is characterizable by a densely ordered structure and λ is the least cardinal such that κλ>κ, then κλ is also characterizable (Theorem 5.4) and (2) if ℵα and κℵα are characterizable cardinals, then the same is true for κℵα+β, for all countable β (Theorem 5.5).Combining these two theorems we get that if κ>2ℵα is a characterizable cardinal, ℵα is characterizable by a densely ordered structure and ℵα is the least cardinal such that κℵα>κ, then for all β<α+ω1, κℵβ is characterizable (Theorem 5.7). Also if κ is a characterizable cardinal, then κℵα is characterizable, for all countable α (Corollary 5.6). This answers a question of the author in Souldatos (submitted for publication) [8]. 相似文献
8.
Donald Mills 《Journal of Number Theory》2002,92(1):87-98
Let Fq denote the finite field of q elements, q=pe odd, let χ1 denote the canonical additive character of Fq where χ1(c)=e2πiTr(c)/p for all c∈Fq, and let Tr represent the trace function from Fq to Fp. We are interested in evaluating Weil sums of the form S=S(a1, …, an)=∑x∈Fq χ1(D(x)) where D(x)=∑ni=1 aixpαi+pβi, αi?βi for each i, is known as a Dembowski-Ostrom polynomial (or as a D-O polynomial). Coulter has determined the value of S when D(x)=axpα+1; in this note we show how Coulter's methods can be generalized to determine the absolute value of S for any D-O polynomial. When e is even, we give a subclass of D-O polynomials whose Weil sums are real-valued, and in certain cases we are able to resolve the sign of S. We conclude by showing how Coulter's work for the monomial case can be used to determine a lower bound on the number of Flq-solutions to the diagonal-type equation ∑li=1 xpγ+1i+(xi+λ)pγ+1=0, where l is even, e/gcd(γ, e) is odd, and h (X)=λpe−γXpe−γ+λpγX is a permutation polynomial over Fq. 相似文献
9.
Let F be a field and let m and n be integers with m,n?3. Let Mn denote the algebra of n×n matrices over F. In this note, we characterize mappings ψ:Mn→Mm that satisfy one of the following conditions:
- 1.
- |F|=2 or |F|>n+1, and ψ(adj(A+αB))=adj(ψ(A)+αψ(B)) for all A,B∈Mn and α∈F with ψ(In)≠0.
- 2.
- ψ is surjective and ψ(adj(A-B))=adj(ψ(A)-ψ(B)) for every A,B∈Mn.
10.
Susana Furtado 《Linear algebra and its applications》2008,429(7):1663-1678
Let F be a field. In [Djokovic, Product of two involutions, Arch. Math. 18 (1967) 582-584] it was proved that a matrix A∈Fn×n can be written as A=BC, for some involutions B,C∈Fn×n, if and only if A is similar to A-1. In this paper we describe the possible eigenvalues of the matrices B and C.As a consequence, in case charF≠2, we describe the possible similarity classes of (P11⊕P22)P-1, when the nonsingular matrix P=[Pij]∈Fn×n, i,j∈{1,2} and P11∈Fs×s, varies.When F is an algebraically closed field and charF≠2, we also describe the possible similarity classes of [Aij]∈Fn×n, i,j∈{1,2}, when A11 and A22 are square zero matrices and A12 and A21 vary. 相似文献
11.
F.S. Cater 《Linear algebra and its applications》1978,19(3):251-274
An n by n matrix M over a (commutative) field F is said to be central if M ? I has rank 1. We say that M is an involution if M2=I; if M is also central we call M a simple involution. We will prove that any n-by-n matrix M satisfying detM=±1 is the product of n+2 or fewer simple involutions. This can be reduced to n+1 if F contains no roots of the equation xn=(?1)n other than ±1. Any ordered field is of this kind. Our main result is that if M is any n-by-n nonsingular nonscalar matrix and if xi ∈ F such that x1?xn=detM, then there exist central matrices Mi such that M=M1?Mn and xi=detMi for i=1,…,n. We will apply this result to the projective group PGL(n,F) and to the little projective group PSL(n,F). 相似文献
12.
Carl Pomerance 《Journal of Combinatorial Theory, Series A》1980,28(2):140-149
Szemerédi's theorem states that given any positive number B and natural number k, there is a number n(k, B) such that if n ? n(k, B) and 0 < a1 < … < an is a sequence of integers with an ? Bn, then some k of the ai form an arithmetic progression. We prove that given any B and k, there is a number m(k, B) such that if m ? m(k, B) and u0, u1, …, um is a sequence of plane lattice points with ∑i=1m…ui ? ui?1… ? Bm, then some k of the ui are collinear. Our result, while similar to Szemerédi's theorem, does not appear to imply it, nor does Szemerédi's theorem appear to imply our result. 相似文献
13.
Peter Borg 《Journal of Combinatorial Theory, Series A》2012,119(4):871-881
Families A1,A2,…,Ak of sets are said to be cross-intersecting if for any i and j in {1,2,…,k} with i≠j, any set in Ai intersects any set in Aj. For a finite set X, let X2 denote the power set of X (the family of all subsets of X). A family H is said to be hereditary if all subsets of any set in H are in H; so H is hereditary if and only if it is a union of power sets. We conjecture that for any non-empty hereditary sub-family H≠{∅} of X2 and any k?|X|+1, both the sum and the product of sizes of k cross-intersecting sub-families A1,A2,…,Ak (not necessarily distinct or non-empty) of H are maxima if A1=A2=?=Ak=S for some largest starSofH (a sub-family of H whose sets have a common element). We prove this for the case when H is compressed with respect to an element x of X, and for this purpose we establish new properties of the usual compression operation. As we will show, for the sum, the condition k?|X|+1 is sharp. However, for the product, we actually conjecture that the configuration A1=A2=?=Ak=S is optimal for any hereditary H and any k?2, and we prove this for a special case. 相似文献
14.
We say that a matrix R∈Cn×n is k-involutary if its minimal polynomial is xk-1 for some k?2, so Rk-1=R-1 and the eigenvalues of R are 1,ζ,ζ2,…,ζk-1, where ζ=e2πi/k. Let α,μ∈{0,1,…,k-1}. If R∈Cm×m, A∈Cm×n, S∈Cn×n and R and S are k-involutory, we say that A is (R,S,μ)-symmetric if RAS-1=ζμA, and A is (R,S,α,μ)-symmetric if RAS-α=ζμA.Let L be the class of m×n(R,S,μ)-symmetric matrices or the class of m×n(R,S,α,μ)-symmetric matrices. Given X∈Cn×t and B∈Cm×t, we characterize the matrices A in L that minimize ‖AX-B‖ (Frobenius norm), and, given an arbitrary W∈Cm×n, we find the unique matrix A∈L that minimizes both ‖AX-B‖ and ‖A-W‖. We also obtain necessary and sufficient conditions for existence of A∈L such that AX=B, and, assuming that the conditions are satisfied, characterize the set of all such A. 相似文献
15.
Let F be any field and let B a matrix of Fq×p. Zaballa found necessary and sufficient conditions for the existence of a matrix A=[Aij]i,j∈{1,2}∈F(p+q)×(p+q) with prescribed similarity class and such that A21=B. In an earlier paper [A. Borobia, R. Canogar, Constructing matrices with prescribed off-diagonal submatrix and invariant polynomials, Linear Algebra Appl. 424 (2007) 615-633] we obtained, for fields of characteristic different from 2, a finite step algorithm to construct A when it exists. In this short note we extend the algorithm to any field. 相似文献
16.
A conjecture of Fraenkel asserts that any partition of the positive integers into m ? 3 sets {⌊αin + βi⌊}n=1∞ - Beatty sequences - with real constants ai and βi, and α1 > α2 > … > αm > 1 satisfies
17.
Let Rij be a given set of μi× μj matrices for i, j=1,…, n and |i?j| ?m, where 0?m?n?1. Necessary and sufficient conditions are established for the existence and uniqueness of an invertible block matrix =[Fij], i,j=1,…, n, such that Fij=Rij for |i?j|?m, F admits either a left or right block triangular factorization, and (F?1)ij=0 for |i?j|>m. The well-known conditions for an invertible block matrix to admit a block triangular factorization emerge for the particular choice m=n?1. The special case in which the given Rij are positive definite (in an appropriate sense) is explored in detail, and an inequality which corresponds to Burg's maximal entropy inequality in the theory of covariance extension is deduced. The block Toeplitz case is also studied. 相似文献
18.
In this paper we confirm a conjecture of Sun which states that each positive integer is a sum of a square, an odd square and a triangular number. Given any positive integer m, we show that p=2m+1 is a prime congruent to 3 modulo 4 if and only if Tm=m(m+1)/2 cannot be expressed as a sum of two odd squares and a triangular number, i.e., p2=x2+8(y2+z2) for no odd integers x,y,z. We also show that a positive integer cannot be written as a sum of an odd square and two triangular numbers if and only if it is of the form 2Tm(m>0) with 2m+1 having no prime divisor congruent to 3 modulo 4. 相似文献
19.
20.
Laurence Boxer 《Topology and its Applications》1980,11(1):17-29
Let X be a finite-dimensional compactum. Let (X) and (X) be the spaces of retractions and non-deformation retractions of X, respectively, with the compact-open (=sup-metric) topology. Let 2Xh be the space of non-empty compact ANR subsets of X with topology induced by the homotopy metric. Let RXh be the subspace of 2Xh consisting of the ANR's in X that are retracts of X.We show that (Sm) is simply-connected for m > 1. We show that if X is an ANR and A0?RXh, then limi→∞Ai=A0 in 2Xh if and only if for every retraction r0 of X onto A0 there are, for almost all i, retractions ri of X onto Ai such that limi→∞ri=ro in (X). We show that if X is an ANR, then the local connectedness of (X) implies that of RXh. We prove that (M) is locally connected if M is a closed surface. We give examples to show how some of our results weaken when X is not assumed to be an ANR. 相似文献