On the duals of the stable rank 3 reflexive sheaves on P3 withc 3 maximal

Summary One proves some properties of the spectrum of the duals of the stable rank 3 reflexive sheaves onP ³ withc ₁=0 andc ₃=c ₂ ² -c ₂ and one determines the Chern classes of these duals.

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Riassunto Si prova alcune proprietà dello spettro degli duali degli fasci riflessivi stabili di rango 3 suP ³ conc ₁=0 ec ₃=c ₂ ² -c ₂ e si determina le classi di Chern di questi duali.

     

Dependence of the wave-front speeds on the propagation direction

The dependence of the characteristic speeds of quasilinear hyperbolic systems on the propagation directionn is investigated. It is proved that any non-vanishing characteristic speedc(n) is the sum of a homogeneous functionc ₁ (n) and a positively homogeneous functionc ₂ (n). As a further result, ifc ₂ (n) is non-vanishing, then bothc ₂ (n)±c ₂ (n) are characteristic speeds.

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Sommario Nel lavoro si analizza la dipendenza dalla direzione di propagazionen delle velocità caratteristiche associate ad un sistema iperbolico quasi lineare. Si prova che ogni velocità caratteristicac(n) non nulla è somma di una funzione omogeneac ₁ (n) e di una funzione positivamente omegeneac ₂(n). Come ulteriore risultato si ha che, sec ₂(n) è non nulla, allora entrambe le funzionic ₁(n)±c ₂(n) sono velocità caratteristiche.

     

Note on the reconstruction of infinite graphs with a fixed finite number of components

For every positive integer c, we construct a pair G_c, H_c of infinite, nonisomorphic graphs both having exactly c components such that G_c and H_c are hypomorphic, i.e., G_c and H_c have the same families of vertex-deleted subgraphs. This solves a problem of Bondy and Hemminger. Furthermore, the pair G₁, H₁ is an example for a pair of non-isomorphic, hypomorphic, connected graphs also having connected complements—a property not shared by any of the previously known counterexamples to the reconstruction conjecture for infinite graphs.     

An algorithm for the determination of longest distances in a graph

This paper presents an algorithm for ranking the vertices of a directed graph. Its space and time requirements are bounded byc ₁ n ² +c ₂, wheren is the number of vertices of the graph andc ₁,c ₂ are positive constants which are independent of the size or other properties of the graph.The algorithm can be easily modified to solve the problem of determining longest distances from a vertex to all other vertices in a positive real valued graph with at mostc ₁ n ² +c ₂ elementary operations; the same result holds for shortest distances in negative real valued graphs.     

On a Maximum Principle for Bergman Spaces with Small Exponents

We prove a stability theorem for the nullity of a linear combination c ₁ P ₁ + c ₂ P ₂ of two idempotent operators P ₁, P ₂ on a Banach space provided c ₁, c ₂ and c ₁ + c ₂ are nonzero. We then show that for c ₁ P ₁ + c ₂ P ₂ the property of being upper semi-Fredholm, lower semi-Fredholm and Fredholm, respectively, is independent of the choice of c ₁, c ₂, and that the nullity, defect and index of c ₁ P ₁ + c ₂ P ₂ are stable.     

On the smallest limited snake of unit disks

A limited snake of size n is a set of nonoverlapping unit disks D ₁, ..., D _nwith centers c ₁, ..., c _nwhere the distances ¦c _i c _j¦=2 if and only if ¦i–j¦=1, and no disk can touch D ₁ or D _nwithout further common points with D ₁, ..., D _nThe size of the smallest limited snake is proved to be 10.     

An Arc as the Inverse Limit of a Single Bonding Map of Type N on [0,1]

Let I = [0, 1], c ₁, c ₂ ∈ (0, 1) with c ₁ < c ₂ and f : I⊸I be a continuous map satisfying: are both strictly increasing and is strictly decreasing. Let A = {x ∈ [0, c ₁]∣f(x) = x}, a=max A, a ₁ =max(A\{a}), and B = {x ∈ [c ₂, 1]∣f(x) = x}, b=minB, b ₁ =min(B\{b}). Then the inverse limit (I, f) is an arc if and only if one of the following three conditions holds: (1) If c ₁ < f (c ₁) ≤ c ₂ (resp. c ₁ ≤ f (c ₂) < c ₂), then f has a single fixed point, a period two orbit, but no points of period greater than two or f has more than one fixed point but no points of other periods, furthermore, if A≠φ and B≠φ, then f (c ₂) > a (resp. f (c ₁) < b). (2) If f (c ₁) ≤ c ₁ (resp. f (c ₂) ≥ c ₂), then f has more than one fixed point, furthermore, if B≠φ and A\ {a} ≠φ, f (c ₂) ≥ a or if a ₁ < f (c ₂) < a, f ² (c ₂) > f (c ₂), (resp. f has more than one fixed point, furthermore, if A≠φ and B\{b}≠φ, f (c ₁) ≤ b or if b < f (c ₂) < b ₁, f ² (c ₁) < f (c ₁)). (3) If f (c ₁) > c ₂ and f (c ₂) < c ₁, then f has a single fixed point, a single period two orbit lying in I\(u, v) but no points of period greater than two, where u, v ∈ [c ₁, c ₂] such that f (u) = c ₂ and f (v) = c ₁. Supported by the National Natural Science Foundation of China (No. 19961001, No. 60334020) and Outstanding Young Scientist Research Fund. (No. 60125310)     

Birational properties of moduli spaces of stable locally free rank-2 sheaves on algebraic surfaces

LetX be a smooth algebraic surface over the complex number field. Fix a polarizationL, an invertible sheafc ₁ and an integerc ₂ such that (4c ₂-c ₁ ² ) is positive. letM _L(c ₁,c ₂) be the moduli space ofL-stable locally free rank-2 sheaves onX with chern classesc ₁ andc ₂ respectively, and let ξ be a numerical equivalence class defining a nonempty wall of type (c ₁,c ₂). We study the properties ofE _ξ(c ₁,c ₂) and obtain estimations for its dimension. Then, we discuss the existence of trivial polarizations, and determine the birational structures of moduli spacesM _L(c ₁,c ₂) whenX is a minimal surface of general type and (4c ₂-c ₁ ² ) is sufficiently large.     

Every order is the endomorphism ring of a projective module over a quasi–hereditary order

We give in this note some new bounds on p _c(G) in terms of σ _c for several classes of finite groups, in particular, we prove that ρ _c ≤4σ _c (G) for any finite solvable group G, which improves some known results. We also pose some related open problems.     

The eigenvalues of the product of matrices with prescribed similarity classes *

Let A and B be n × n nonsingular matrices over a field F, and c ₁,…,c _n ∈ F. We give a necessary and sufficient condition for the existence of matrices A′ and B′ similar to A and B, respectively, such that A′ B′ has eigenvalues c ₁,…,c _n.     

A phase transition in the random transposition random walk

Our work is motivated by Bourque and Pevzner's (2002) simulation study of the effectiveness of the parsimony method in studying genome rearrangement, and leads to a surprising result about the random transposition walk on the group of permutations on n elements. Consider this walk in continuous time starting at the identity and let D _t be the minimum number of transpositions needed to go back to the identity from the location at time t. D _t undergoes a phase transition: the distance D _{cn /2}∼u(c)n, where u is an explicit function satisfying u(c)=c/2 for c≤1 and u(c)<c/2 for c>1. In addition, we describe the fluctuations of D _{cn /2} about its mean in each of the three regimes (subcritical, critical and supercritical). The techniques used involve viewing the cycles in the random permutation as a coagulation-fragmentation process and relating the behavior to the Erdős-Renyi random graph model.     

A refined Schwarz inequality on the boundary

We consider a function f holomorphic in the unit disc D with f(D)???D and f(0)?=?f(z ₀)?=?0, for 0?<?|z ₀|?<?1. We obtain sharp lower bounds on the angular derivative f′(c) at the point c where |c|?=?|f(c)|?=?1.     

An Outer Commutator Multiplier and Capability of Finitely Generated Abelian Groups

We present an explicit structure for the Baer invariant of a finitely generated abelian group with respect to the variety [𝔑 _{c 1} , 𝔑 _{c 2} ], for all c ₂ ≤ c ₁ ≤ 2c ₂. As a consequence, we determine necessary and sufficient conditions for such groups to be [𝔑 _{c 1} , 𝔑 _{c 2} ]-capable. We also show that if c ₁ ≠ 1 ≠ c ₂, then a finitely generated abelian group is [𝔑 _{c 1} , 𝔑 _{c 2} ]-capable if and only if it is capable. Finally, we show that 𝔖₂-capability implies capability, but there is a capable finitely generated abelian group which is not 𝔖₂-capable.     

Measures on phase space as solutions of the one-dimensional neutron transport equation

Summary The linear integral transport operator for slab geometry is formulated and studied as a mapping on the set of measures on the phase space of the underlying system, with the expected number of neutrons emergent from a collision represented by a measure on the space of outgoing velocities. Under appropriate assumptions it is shown that, if c represents the maximum number of secondary particles per collision, then there exists c ₁ ≥1 such that the system is subcritical for c≤c ₁ . An example shows that c ₁ ≥1 is sharp in general, but further assumptions are given under which one can deduce c ₁ >1. The idealized laws of elastic and inelastic scattering are shown to satisfy our assumptions. Entrata in Redazione il 27 ottobre 1975.     

The eigenvalues of the sum of matrices with prescribed invariant polynomials *

Let A and B be n×n matrices over a field F, and c ₁,…,c_n ∈F. We give a sufficient condition for the existence of matrices A' and B' similar to A and B, respectively, such that A' + B' has eigenvalues c ₁,…,c_n .     

On the automatic derivation of a set of geometric formulae

Leta, b, andc be the three sides of a triangleABC, a _i ,b _i ,c _i anda _e ,b _e , c_e be the lengths of the three internal and external bisectors of the three anglesA, B, andC respectively. It is easy to express the bisectors as formulae of the sides. In this paper, we solve a problem proposed by H. Zassenhaus: for any three different bisectors in {a_i, b_i, c_i, a_e, b_e, c_e}, finding the relations between each side of the triangle and the three chosen bisectors. We also prove that given any general values for three different bisectors (internal or external) of a triangle, we can not draw the triangle using a ruler and a pair of compasses alone. The formulae mentioned above are derived automatically using a general method of mechanical formula derivation.This work was partially supported by a Grant from Chinese NSF and by the NSF Grant CCR-917870.     

On the Erd s-Ginzburg-Ziv theorem and the Ramsey numbers for stars and matchings

A link between Ramsey numbers for stars and matchings and the Erd s-Ginzburg-Ziv theorem is established. Known results are generalized. Among other results we prove the following two theorems. Theorem 5. Let m be an even integer. If c : e (K_2m−1)→{0, 1,…, m−1} is a mapping of the edges of the complete graph on 2m−1 vertices into {0, 1,…, m−1}, then there exists a star K_1,m in K_2m−1 with edges e₁, e₂,…, e_m such that c(e₁)+c(e₂)++c(e_m)≡0 (mod m). Theorem 8. Let m be an integer. If c : e(K^r_(r+1)m−1)→{0, 1,…, m−1} is a mapping of all the r-subsets of an (r+1)m−1 element set S into {0, 1,…, m−1}, then there are m pairwise disjoint r-subsets Z₁, Z₂,…, Z_m of S such that c(Z₁)+c(Z₂)++c(Z_m)≡0 (mod m).     

The invertibility of the difference and the sum of commuting generalized and hypergeneralized projectors

In this article, we study necessary and sufficient conditions for the invertibility of a linear combination c ₁ A ^k ?+?c ₂ B ^l , in the case when A and B are both commuting generalized or hypergeneralized projectors. We present some results relating different matrix partial orderings and the invertibility of a linear combination c ₁ A ^k ?+?c ₂ B ^l when A and B are hypergeneralized projectors.     

Fractional and integral colourings

LetG=(V, E) be an undirected graph andc any vector in ℤ ^V(G) ₊. Denote byχ(G _c) (resp.η(G _c)) the chromatic number (resp. fractional chromatic number) ofG with respect toc. We study graphs for whichχ(G _c)−[η(G _c)]⩽1. We show that for the class of graphs satisfyingχ(G _c)=[η(G _c)] (a class generalizing perfect graphs), an analogue of the Duplication Lemma does not hold. We also describe a 2-vertex cut decomposition procedure related to the integer decomposition property. We use this procedure to show thatχ(G _c)=[η(G _c)] for series-parallel graphs andχ(G _c)⩽[η(G _c)]+1 for graphs that do not have the 4-wheel as a minor. The work of this author was supported by the Natural Sciences and Engineering Research Council of Canada (NSERCC) under grant A9126.     

On nonsingularity of combinations of three group invertible matrices and three tripotent matrices

Let T?=?c ₁ T ₁?+?c ₂ T ₂?+?c ₃ T ₃???c ₄(T ₁ T ₂?+?T ₃ T ₁?+?T ₂ T ₃), where T ₁, T ₂, T ₃ are three n?×?n tripotent matrices and c ₁, c ₂, c ₃, c ₄ are complex numbers with c ₁, c ₂, c ₃ nonzero. In this article, necessary and sufficient conditions for the nonsingularity of such combinations are established and some formulae for the inverses of them are obtained. Some of these results are given in terms of group invertible matrices.