Packing nearly-disjoint sets

De Bruijn and Erdős proved that ifA ₁, ...,A _k are distinct subsets of a set of cardinalityn, and |A _i ∩A _j |≦1 for 1≦i<j ≦k, andk>n, then some two ofA ₁, ...,A _k have empty intersection. We prove a strengthening, that at leastk /n ofA ₁, ...,A _k are pairwise disjoint. This is motivated by a well-known conjecture of Erdőds, Faber and Lovász of which it is a corollary. Partially supported by N. S. F. grant No. MCS—8103440     

On P.I. ring and standard identities

An example is given of a ringR (with 1) satisfying the standard identityS ₆[x ₁, ...,x ₆] butM ₂(R), the 2 × 2 matrix ring overR, does not satisfyS ₁₂[x ₁, ...,x ₁₂]. This is in contrast to the caseR=M _n (F),F a field, where by the Amitsur-Levitzki theoremR satisfiesS _2n [x ₁, ...,x _2n] andM ₂(R) satisfiesS _4n [x ₁, ...,x _n]. Part of this work was done while the author enjoyed the hospitality of the University of California at San Diego, the University of Texas at Austin and the University of Washington at Seattle.     

On a recursive inverse eigenvalue problem

Let s ₁, ..., s _n be arbitrary complex scalars. It is required to construct an n × n normal matrix A such that s _i is an eigenvalue of the leading principal submatrix A _i , i = 1, 2, ..., n. It is shown that, along with the obvious diagonal solution diag(s ₁, ..., s _n ), this problem always admits a much more interesting nondiagonal solution A. As a rule, this solution is a dense matrix; with the diagonal solution, it shares the property that each submatrix A _i is itself a normal matrix, which implies interesting connections between the spectra of the neighboring submatrices A _i and A _{i + 1}.     

Hermite Interpolation and Sobolev Orthogonality

In this paper, we study orthogonal polynomials with respect to the bilinear form (f, g) _S = V(f) A V(g) ^T + <u, f ^(N) g ^(N)V(f) =(f(c ₀), f "(c ₀), ..., f ^{(n – 1)} ₀(c ₀), ..., f(c _p ), f "(c _p ), ..., f ^{(n – 1)} _p(c _p )) u is a regular linear functional on the linear space P of real polynomials, c ₀, c ₁, ..., c _p are distinct real numbers, n ₀, n ₁, ..., n _p are positive integer numbers, N=n ₀+n ₁+...+n _p , and A is a N × N real matrix with all its principal submatrices nonsingular. We establish relations with the theory of interpolation and approximation.     

Relative MilnorK-theory

LetR be a commutative, semi-local ring,I ₁, ...,I _s ideals. In this paper, we define therelative Milnor K-groups of (R;I ₁, ...,I _s ),K _p ^M (R;I ₁, ...,I _s ), and show that these groups have many of the properties of the usual MilnorK-groups of a field. In particular, assuming a weak condition on the ideals, we show thatK _p ^M (R;I ₁, ...,I _s ) is isomorphic to the weightp portion of the relative QuillenK-groupK _p (R;I ₁, ...,I _s ), after inverting (p–1)!. We also define the relative group homology of GL _n (R;I ₁, ...,I _s ), and show thatK _p ^M (R;I ₁, ...,I _s ) is isomorphic toH _p (GL_p(R;I ₁, ...,I _s ))/Im(H _p (GL _p–1 (R;I ₁, ...,I _s ))). Finally, we consider a generalization to the relative setting of Kato's conjecture asserting that the Galois symbol gives an isomorphism fromK _p ^M (F)/l ^v to , and show that this relative version of Kato's conjecture implies the Quillen-Lichtenbaum conjectures asserting the Chern class:

Cyclic homology and the Macdonald conjectures

Summary LetA+(k) denote the ring [t]/t ^k+1 and letG be a reductive complex Lie algebra with exponentsm ₁, ...,m n. This paper concerns the Lie algebra cohomology ofGA ₊(k) considered as a bigraded algebra (here one of the gradings is homological degree and the other, which we callweight, is inherited from the obvious grading ofGA ₊(k)). We conjecture that this Lie algebra cohomology is an exterior algebra withk+1 generators of homological degree 2m _s +1 fors=1,2, ...,n. Of thesek+1 generators of degree 2m _s +1, one has weight 0 and the others have weights (k+1)m _s +t fort=1,2, ...,k.It is shown that this conjecture about the Lie algebra cohomology of A ₊(k) implies the Macdonald root system conjectures. Next we consider the case thatG is a classical Lie algebra with root systemA _n ,B _n ,C _n , orD _n. It is shown that our conjecture holds in the limit onn asn approaches infinity which amounts to the computation of the cyclic and dihedral cohomologies ofA+(k). Lastly we discuss the relevance of this limiting case to the case of finiten in this situation.Partially supported by NSF grant number MCS-8401718 and a Bantrell Fellowship     

Lipschitz functions,Schatten ideals,and unbounded derivations

It is proved that, for any Lipschitz function f(t ₁, ..., t _n ) of n variables, the corresponding map f _op: (A ₁, ...,A _n ) → f(A ₁, ..., A _n ) on the set of all commutative n-tuples of Hermitian operators on a Hilbert space is Lipschitz with respect to the norm of each Schatten ideal S ^p , p ∈ (1,∞). This result is applied to the functional calculus of normal operators and contractions. It is shown that Lipschitz functions of one variable preserve domains of closed derivations with values in S ^p . It is also proved that the map f _op is Fréchet differentiable in the norm of S ^p if f is continuously differentiable.     

Renormalization of multiple <Emphasis Type="Italic">q</Emphasis>-zeta values

In this paper, we shall define the renormalization of the multiple q-zeta values （MqZV） which are special values of multiple q-zeta functions ζq （s1,..., Sd） when the arguments are all positive integers or all non-positive integers. This generalizes the work of Guo and Zhang （Renormalization of Multiple Zeta Values, arxiv： math/0606076v3）. We show that our renormalization process produces the same values if the MqZVs are well-defined originally and that these renormalizations of MqZV satisfy the q-stuffle relations if we use shifted-renormalizations for all divergent ζq（S1,..., sd） （i.e., s1 ≤ 1）. Moreover, when q ↑ 1 our renormalizations agree with those of Guo and Zhang.     

The Hadwiger Number of Jordan Regions Is Unbounded

We show that for every n > 0 there is a planar topological disk A₀ and n translates A₁, A₂, ..., A_n of A₀ such that the interiors of A₀, ..., A_n are pairwise disjoint, but with each A_i touching A₀ for 1 ≤ i ≤ n.     

Necessary and sufficient condition in Lyapunov robust control: Multi-input case

We consider a linear time-invariant finite-dimensional system x=Ax+Bu with multi-inputu, in which the matricesA andB are in canonical controller form. We assume that the system is controllable andB has rankm. We study the Lyapunov equationPA+A ^T P+Q=0, withQ>0, and investigate the properties thatP must satisfy in order that the canonical controller matrixA be Hurwitz. We show that, for the matrixA being Hurwitz, it is necessary and sufficient thatB ^T PB>0 and that the determinant ofB ^T PW be Hurwitz, whereW=block diag[w ₁,...,w _m ], with elementw _i =[s ^k _i –1,s ^k _i –2,...,s, 1] ^T ; here, the symbolsk _i ,i=1, 2, ...,m, denote the Kronecker invariants with respect to the pair {A, B}. This result has application in designing robust controllers for linear uncertain systems.     

The strong approximation of extremal processes

Summary If X ₁, X ₂, ..., are i.i.d. random variables and Y _n =Max(X ₁, ..., X _n ); if for some sequences A _n , B_n, n=1, 2, ..., E _n (t)=A_nY_[nt]+B_n is such that E _n (1) weakly converges to a non degenerate limit distribution, then we prove that it is possible to construct a sequence of replicates of extremal processes E ⁽ⁿ⁾(t) on the same probability space, such that d(E _n (.), E ⁽ⁿ⁾(.))0 a.s., with the Levy metric. We give the rates of consistency of the approximations.     

Self-orthogonal cyclicn-quasigroups

A quasigroup (Q,) satisfying the identityx(yx) =y (or the equivalent identity (xy)x =y) is called semisymmetric. Ann-quasigroup (Q, A) satisfying the identityA(A(x ₁, ...,x _n ),x ₁, ...,x _n–1) =x _n is called cyclic. So, cyclicn-quasigroups are a generalization of semisymmetric quasigroups. In this paper, self-orthogonal cyclicn-quasigroups (SOCnQs) are considered. Some constructions ofSOCnQs are described and the spectrum of suchn-quasigroups investigated.     

Density Modulo 1 of Sublacunary Sequences

We prove the existence of real numbers badly approximated by rational fractions whose denominators form a sublacunar sequence. For example, for the ascending sequence s _n , n = 1, 2, 3, ..., generated by the ordered numbers of the form 2ⁱ3^j, i, j = 1, 2, 3, ..., we prove that the set of real numbers α such that inf _n∈ℕ n‖s _n α‖ > 0 is a set of Hausdorff dimension 1. The divergence of the series implies that the Lebesgue measure of those numbers is zero.__________Translated from Matematicheskie Zametki, vol. 77, no. 6, 2005, pp. 803–813.Original Russian Text Copyright ©2005 by R. K. Akhunzhanov, N. G. Moshchevitin.     

A direct method for the general solution of a system of linear equations

A computationally stable method for the general solution of a system of linear equations is given. The system isA ^Tx–B=0, where then-vectorx is unknown and then×q matrixA and theq-vectorB are known. It is assumed that the matrixA ^T and the augmented matrix [A ^T,B] are of the same rankm, wheremn, so that the system is consistent and solvable. Whenm<n, the method yields the minimum modulus solutionx _m and a symmetricn ×n matrixH _m of rankn–m, so thatx=x _m+H _my satisfies the system for ally, ann-vector. Whenm=n, the matrixH _m reduces to zero andx _m becomes the unique solution of the system.The method is also suitable for the solution of a determined system ofn linear equations. When then×n coefficient matrix is ill-conditioned, the method can produce a good solution, while the commonly used elimination method fails.This research was supported by the National Science Foundation, Grant No. GP-41158.     

Applications of BGP-reflection functors: isomorphisms of cluster algebras

Given a symmetrizable generalized Cartan matrix A, for any index k, one can define an automorphism associated with A, of the field Q(u1,…, un) of rational functions of n independent indeterminates u1,…,un.It is an isomorphism between two cluster algebras associated to the matrix A (see sec. 4 for the precise meaning). When A is of finite type, these isomorphisms behave nicely; they are compatible with the BGP-reflection functors of cluster categories defined in a previous work if we identify the indecomposable objects in the categories with cluster variables of the corresponding cluster algebras, and they are also compatible with the "truncated simple reflections" defined by Fomin-Zelevinsky. Using the construction of preprojective or preinjective modules of hereditary algebras by DIab-Ringel and the Coxeter automorphisms (i.e. a product of these isomorphisms), we construct infinitely many cluster variables for cluster algebras of infinite type and all cluster variables for finite types.     

Generalized perfect arrays and menon difference sets

Given an s ₁ × ... × s _rinteger-valued array A and a (0, 1) vector z = (z ₁, ..., z _r), form the array A from A by recursively adjoining a negative copy of the current array for each dimension i where z _i = 1. A is a generalized perfect array type z if all periodic autocorrelation coefficients of A are zero, except for shifts (u ₁, ..., u _r) where u _i, - 0 (mod s _i) for all i. The array is perfect if z = (0, ..., 0) and binary if the array elements are all ±1. A nontrivial perfect binary array (PBA) is equivalent to a Menon difference set in an abelian group.Using only elementary techniques, we prove various construction theorems for generalized perfect arrays and establish conditions on their existence. We show that a generalized PBA whose type is not (0, ..., 0) is equivalent to a relative difference set in an abelian factor group. We recursively construct several infinite families of generalized PBAs, and deduce nonexistence results for generalized PBAs whose type is not (0, ..., 0) from well-known nonexistence results for PBAs. A central result is that a PBA with 2^2y 3^2u elements and no dimension divisible by 9 exists if and only if no dimension is divisible by 2 ^y+2. The results presented here include and enlarge the set of sizes of all previously known generalized PBAs.     

Simultaneous similarity of matrices

In this paper we solve completely and explicitly the long-standing problem of classifying pairs of n × n complex matrices (A, B) under the simultaneous similarity (TAT⁻¹, TBT⁻¹). Roughly speaking, the classification decomposes to a finite number of steps. In each step we consider an open algebraic set ⁰_n,2,r,π M_n × M_n (M_n = the set of n × n complex-valued matrices). Here r and π are two positive integers. Then we construct a finite number of rational functions ø₁,…,ø_s in the entries of A and B whose values are constant on all pairs similar in _n,2,r,π to (A, B). The values of the functions ø_i(A, B), I = 1,…, s, determine a finite number (at most κ(n, 2, r)) of similarity classes in _n,2,r,π. Let S_n be the subspace of complex symmetric matrices in M_n. For (A, B) ε S_n × S_n we consider the similarity class (TAT^t, TBT^t), where T ranges over all complex orthogonal matrices. Then the characteristic polynomial |λI − (A + xB)| determines a finite number of similarity classes for almost all pairs (A, B) ε S_n × S_n.     

A problem related to Foulkes’s conjecture

In this paper we study a class of symmetric matricesT indexed by positive integers m≥ n≥2 and defined as follows: for any positive integersp andq let ?_p,q be the set of partitions ofU = {1,2,3, ...,pq} into p blocks each of sizeq. Letm ≥n ≥ 2 be positive integers. By atransversal of α = A₁/A₂/.../A_n ∈ ?_n,m we mean a partitionß = B₁/B₂/.../B_m ∈ ? _m,n such that ‖A _i ∩B _j = 1 for every i= 1,2, ...,n and everyj = 1,2, ...,m. LetM be the zero-one matrix with rows indexed by the elements of ?_n,m and columns indexed by the elements of ?_m,n such that M_αß = 1 iffß is a transversal of α. We are interested in finding the eigenvalues and eigenspaces of the symmetric matrixT = MM^t. The nonsingularity ofT implies Foulkes’s Conjecture (for these values of m andn). In the casen = 2 we completely determine the eigenvalues and eigenspaces of T and in so doing demonstrate the non-singularity ofT. Forn = 3 we develop a fast algorithm for computing the eigenvalues ofT, and give numerical results in the cases m = 3,4, 5, 6.