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1.
When we interpret modal ? as the limit point operator of a topological space, the Gödel-Löb modal system GL defines the class Scat of scattered spaces. We give a partition of Scat into α-slices S α , where α ranges over all ordinals. This provides topological completeness and definability results for extensions of GL. In particular, we axiomatize the modal logic of each ordinal α, thus obtaining a simple proof of the Abashidze–Blass theorem. On the other hand, when we interpret ? as closure in a topological space, the Grzegorczyk modal system Grz defines the class HI of hereditarily irresolvable spaces. We also give a partition of HI into α-slices H α , where α ranges over all ordinals. For a subset A of a hereditarily irresolvable space X and an ordinal α, we introduce the α-representation of A, give an axiomatization of the α-representation of A, and characterize H α in terms of α-representations. We prove that ${X \in {\bf H}_{1}}When we interpret modal ◊ as the limit point operator of a topological space, the G?del-L?b modal system GL defines the class Scat of scattered spaces. We give a partition of Scat into α-slices S
α
, where α ranges over all ordinals. This provides topological completeness and definability results for extensions of GL. In particular, we axiomatize the modal logic of each ordinal α, thus obtaining a simple proof of the Abashidze–Blass theorem. On the other hand, when we interpret ◊ as closure in a topological
space, the Grzegorczyk modal system Grz defines the class HI of hereditarily irresolvable spaces. We also give a partition of HI into α-slices H
α
, where α ranges over all ordinals. For a subset A of a hereditarily irresolvable space X and an ordinal α, we introduce the α-representation of A, give an axiomatization of the α-representation of A, and characterize H
α
in terms of α-representations. We prove that X ? H1{X \in {\bf H}_{1}} iff X is submaximal. For a positive integer n, we generalize the notion of a submaximal space to that of an n-submaximal space, and prove that X ? Hn{X \in {\bf H}_{n}} iff X is n-submaximal. This provides topological completeness and definability results for extensions of Grz. We show that the two partitions are related to each other as follows. For a successor ordinal α = β + n, with β a limit ordinal and n a positive integer, we have Ha ?Scat = Sb+2n-1 èSb+2n{{\bf H}_{\alpha} \cap {\bf Scat} = {\bf S}_{\beta+2n-1} \cup {\bf S}_{\beta+2n}} , and for a limit ordinal α, we have Ha ?Scat = Sa{{\bf H}_{\alpha} \cap {\bf Scat} = {\bf S}_{\alpha}} . As a result, we obtain full and faithful translations of ordinal complete extensions of Grz into ordinal complete extensions of GL, thus generalizing the Kuznetsov–Goldblatt–Boolos theorem. 相似文献
2.
Let Q be an alphabet with q elements. For any code C over Q of length n and for any two codewords a = (a 1, . . . , a n ) and b = (b 1, . . . , b n ) in C, let ${D({\bf a, b}) = \{(x_1, . . . , x_n) \in {Q^n} : {x_i} \in \{a_i, b_i\}\,{\rm for}\,1 \leq i \leq n\}}Let Q be an alphabet with q elements. For any code C over Q of length n and for any two codewords a = (a
1, . . . , a
n
) and b = (b
1, . . . , b
n
) in C, let D(a, b) = {(x1, . . . , xn) ? Qn : xi ? {ai, bi} for 1 £ i £ n}{D({\bf a, b}) = \{(x_1, . . . , x_n) \in {Q^n} : {x_i} \in \{a_i, b_i\}\,{\rm for}\,1 \leq i \leq n\}}. Let C* = èa, b ? CD(a, b){C^* = {{\bigcup}_{\rm {a,\,b}\in{C}}}D({\bf a, b})}. The code C is said to have the identifiable parent property (IPP) if, for any x ? C*{{\rm {\bf x}} \in C^*}, ?x ? D(a, b){a, b} 1 ?{{\bigcap}_{{\rm x}{\in}D({\rm a,\,b})}\{{\bf a, b}\}\neq \emptyset} . Codes with the IPP were introduced by Hollmann et al [J. Combin. Theory Ser. A 82 (1998) 21–133]. Let F(n, q) = max{|C|: C is a q-ary code of length n with the IPP}.T? and Safavi-Naini [SIAM J. Discrete Math. 17 (2004) 548–570] showed that 3q + 6 - 6 é?{q+1}ù £ F(3, q) £ 3q + 6 - é6 ?{q+1}ù{3q + 6 - 6 \lceil\sqrt{q+1}\rceil \leq F(3, q) \leq 3q + 6 - \lceil 6 \sqrt{q+1}\rceil}, and determined F (3, q) precisely when q ≤ 48 or when q can be expressed as r
2 + 2r or r
2 + 3r +2 for r ≥ 2. In this paper, we establish a precise formula of F(3, q) for q ≥ 24. Moreover, we construct IPP codes of size F(3, q) for q ≥ 24 and show that, for any such code C and any x ? C*{{\rm {\bf x}} \in C^*}, one can find, in constant time, a ? C{{\rm {\bf a}} \in C} such that if x ? D (c, d){{\rm {\bf x}} \in D ({\bf c, d})} then a ? {c, d}{{\rm {\bf a}} \in \{{\rm {\bf c, d}}\}}. 相似文献
3.
If G has a nilpotent normal p-complement and V is a finite, faithful and completely reducible G-module of characteristic p, we prove that there exist ${v_1, v_2 \in V}If G has a nilpotent normal p-complement and V is a finite, faithful and completely reducible G-module of characteristic p, we prove that there exist v1, v2 ? V{v_1, v_2 \in V} such that CG(v1)?CG(v2) = P{{\bf C}_{G}{(v_1)}\cap {\bf C}_{G}{(v_2)} = P} , where P ? Sylp(G){P \in {\rm Syl}_p(G)} . We hence deduce that, if the normal p-complement K is nontrivial, there exists v ? CV(P){v \in {\bf C}_{V}(P)} such that |K : C
K
(v)|2 > |K|. 相似文献
4.
In this paper we describe a function F n : R + → R + such that for any hyperbolic n-manifold M with totally geodesic boundary ${\partial M \neq \emptyset}In this paper we describe a function F
n
: R
+ → R
+ such that for any hyperbolic n-manifold M with totally geodesic boundary ?M 1 ?{\partial M \neq \emptyset} , the volume of M is equal to the sum of the values of F
n
on the orthospectrum of M. We derive an integral formula for F
n
in terms of elementary functions. We use this to give a lower bound for the volume of a hyperbolic n-manifold with totally
geodesic boundary in terms of the area of the boundary. 相似文献
5.
Asaf Nachmias 《Geometric And Functional Analysis》2009,19(4):1171-1194
Let {G n } be a sequence of finite transitive graphs with vertex degree d = d(n) and |G n | = n. Denote by p t (v, v) the return probability after t steps of the non-backtracking random walk on G n . We show that if p t (v, v) has quasi-random properties, then critical bond-percolation on G n behaves as it would on a random graph. More precisely, if $\mathop {\rm {lim\, sup\,}} \limits_{n} n^{1/3} \sum\limits_{t = 1}^{n^{1/3}} {t{\bf p}^t(v,v) < \infty ,}$ then the size of the largest component in p-bond-percolation with ${p =\frac{1+O(n^{-1/3})}{d-1}}Let {G
n
} be a sequence of finite transitive graphs with vertex degree d = d(n) and |G
n
| = n. Denote by p
t
(v, v) the return probability after t steps of the non-backtracking random walk on G
n
. We show that if p
t
(v, v) has quasi-random properties, then critical bond-percolation on G
n
behaves as it would on a random graph. More precisely, if
lim sup n n1/3 ?t = 1n1/3 tpt(v,v) < ¥,\mathop {\rm {lim\, sup\,}} \limits_{n} n^{1/3} \sum\limits_{t = 1}^{n^{1/3}} {t{\bf p}^t(v,v) < \infty ,} 相似文献
6.
Peter W. Michor 《Proceedings of the American Mathematical Society》1997,125(7):2175-2177
The assumption in the main result of Basic differential forms for actions of Lie groups (Proc. Amer. Math. Soc. 124 (1996), 1633-1642) is removed.
7.
Let r ∈ N, α, t ∈ R, x ∈ R 2, f: R 2 → C, and denote $ \Delta _{t,\alpha }^r (f,x) = \sum\limits_{k = 0}^r {( - 1)^{r - k} c_r^k f(x_1 + kt\cos \alpha ,x_2 + kt\sin \alpha ).} $ In this paper, we investigate the relation between the behavior of the quantity $ \left\| {\int\limits_E {\Delta _{t,\alpha }^r (f, \cdot )\Psi _n (t)dt} } \right\|_{p,G} , $ as n → ∞ (here, E ? R, G ∈ {R 2, R + 2 }, and ψ n ∈ L 1(E) is a positive kernel) and structural properties of function f. These structural properties are characterized by its “directional” moduli of continuity: $ \omega _{r,\alpha } (f,h)_{p,G} = \mathop {\sup }\limits_{0 \leqslant t \leqslant h} \left\| {\Delta _{t,\alpha }^r (f)} \right\|_{p,G} . $ Here is one of the results obtained. Theorem 1. Let E and A be intervals in R + such that A ? E, f ∈ L p (G), α ∈ [0, 2π] when G =R 2 and α ∈ [0, π/2] when G = R + 2 Denote Δ n, k = ∫ A t k ψ n (t)dt. If there exists an r ∈ N such that, for any m ∈ N, we have Δ m, r > 0, Δ m, r + 1 < ∞, and $ \mathop {\lim }\limits_{n \to \infty } \frac{{\Delta _{n,r + 1} }} {{\Delta _{n,r} }} = 0,\mathop {\lim }\limits_{n \to \infty } \Delta _{n,r}^{ - 1} \int\limits_{E\backslash A} {\Psi _n = 0} , $ then the relations $ \mathop {\lim }\limits_{n \to \infty } \Delta _{n,r}^{ - 1} \left\| {\int\limits_E {\Delta _{t,\alpha }^r (f, \cdot )\Psi _n dt} } \right\|_{p,G} \leqslant K, \mathop {\sup }\limits_{t \in (0,\infty )} t^r \omega _{r,\alpha } (f,t)_{p,G} \leqslant K $ are equivalent. Particular methods of approximation are considered. We establish Corollary 1. Let p, G, α, and f be the same as in Theorem 1, and $ \sigma _{n,\alpha } (f,x) = \frac{2} {{\pi n}}\int\limits_{R_ + } {\Delta _{t,\alpha }^1 (f,x)} \left( {\frac{{\sin \frac{{nt}} {2}}} {t}} \right)^2 dt. $ Then the relations $ \mathop {\underline {\lim } }\limits_{n \to \infty } \frac{{\pi n}} {{\ln n}}\left\| {\sigma _{n,\alpha } (f)} \right\|_{p,G} \leqslant K
8.
We consider the solution of the system of linear algebraic equations which arises from the finite element discretization of boundary value problems associated to the differential operator I. The natural setting for such problems is in the Hilbert space H and the variational formulation is based on the inner product in H. We show how to construct preconditioners for these equations using both domain decomposition and multigrid techniques. These preconditioners are shown to be spectrally equivalent to the inverse of the operator. As a consequence, they may be used to precondition iterative methods so that any given error reduction may be achieved in a finite number of iterations, with the number independent of the mesh discretization. We describe applications of these results to the efficient solution of mixed and least squares finite element approximations of elliptic boundary value problems.
9.
TextThe Bowman–Bradley theorem asserts that the multiple zeta values at the sequences obtained by inserting a fixed number of twos between add up to a rational multiple of a power of π. We establish its counterpart for multiple zeta-star values by showing an identity in a non-commutative polynomial algebra introduced by Hoffman.VideoFor a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=LpqA2OJ6vP8. 相似文献
10.
TextOne of the most important statistics in studying the zeros of L-functions is the 1-level density, which measures the concentration of zeros near the central point. Fouvry and Iwaniec (2003) [FI] proved that the 1-level density for L-functions attached to imaginary quadratic fields agrees with results predicted by random matrix theory. In this paper, we show a similar agreement with random matrix theory occurring in more general sequences of number fields. We first show that the main term agrees with random matrix theory, and similar to all other families studied to date, is independent of the arithmetic of the fields. We then derive the first lower order term of the 1-level density, and see the arithmetic enter.VideoFor a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=zpb-gu3G8i0. 相似文献
11.
Schatten-von Neumann Properties in the Weyl Calculus, and Calculus of Metrics on Symplectic Vector Spaces 总被引:1,自引:0,他引:1
Joachim Toft 《Annals of Global Analysis and Geometry》2006,30(2):169-209
Let s w p be the set of all a ∈ ? such that a w (x, D) is Schatten p-operator on L 2. Then we prove the following:
12.
Let H be the symmetric second-order differential operator on L 2(R) with domain ${C_c^\infty({\bf R})}
13.
14.
We attempt a broad exploration of properties and connections between the symmetry function of a convex set S ${S \subset\mathbb{R}^n}
15.
Claus Scheiderer 《Mathematische Zeitschrift》2010,266(1):1-19
Let A be an excellent local ring of real dimension ≤2, let T be a finitely generated preordering in A, and let ${\widehat{T}}
16.
Atsushi Murase 《Mathematische Annalen》2010,347(3):529-543
Let f be a holomorphic cusp form of weight l on SL2(Z) and Ω an algebraic Hecke character of an imaginary quadratic field K with Ω((α)) = (α/|α|) l for ${\alpha\in K^{\times}}
17.
P. Mark Kayll 《Graphs and Combinatorics》2010,26(5):721-726
König–Egerváry graphs are those whose maximum matchings are equicardinal to their minimum-order coverings by vertices. Edmonds (J Res Nat Bur Standards Sect B 69B:125–130, 1965) characterized the perfect matching polytope of a graph G = (V, E) as the set of nonnegative vectors ${{\bf{x}}\in\mathbb R^E}
18.
19.
Yair Glasner 《Transformation Groups》2009,14(4):787-800
Let A be a locally compact group topologically generated by d elements and let k > d. Consider the action, by precomposition, of Γ = Aut(F k ) on the set of marked, k-generated, dense subgroups $ {D_{k,A}}: = \left\{ {\eta \in {\text{Hom}}\left( {{F_k},A} \right)\left| {\overline {\left\langle {\phi \left( {{F_k}} \right)} \right\rangle } = A} \right.} \right\}
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