Heights of Characters in Blocks of p-Solvable Groups

In this paper, it is proved that if B is a Brauer p-block ofa p-solvable group, for some odd prime p, then the height ofany ordinary character in B is at most 2b, where p^b is the largestdegree of the irreducible characters of the defect group ofB. Some other results that relate the heights of characterswith properties of the defect group are obtained. 2000 MathematicsSubject Classification 20C15, 20C20.     

Generalized Blocks of Unipotent Characters in the Finite General Linear Group

In an article of 2003, Külshammer, Olsson, and Robinson defined ?-blocks for the symmetric groups, where ? > 1 is an arbitrary integer, and proved that they satisfy an analogue of the Nakayama Conjecture. Inspired by this work and the definitions of generalized blocks and sections given by the authors, we give in this article a definition of d-sections in the finite general linear group, and construct d-blocks of unipotent characters, where d ≥ 1 is an arbitrary integer. We prove that they satisfy one direction of an analogue of the Nakayama Conjecture, and, in some cases, prove the other direction. We also prove that they satisfy an analogue of Brauer's Second Main Theorem.     

Separating points by spheres

With respect to a collection of N + m + 1 points in E^m and an integer k, 0 ? k ? N; a criterion is given for the existence of a hypersphere through m specified points and one other, so that the remaining N points are separated into k interior and N ? k exterior. This result is used to solve some particular problems of separating points in the plane by circles.     

More Bounds on Norms of Generalized Characters with Applications to p-Local Bounds and Blocks

A new inequality concerning generalized characters of p-groupsis obtained and applications to bounding the number of irreduciblecharacters in blocks of finite groups are given. 2000 MathematicsSubject Classification 20C20.     

Separating words by occurrences of subwords

We obtain some lower bounds for the complexity of word separation by occurrences of subwords. In the case of length 1 subwords, we show that the bound is exact up to a constant factor. Connection with the problem of separating words by automata is considered.     

Separating convex sets by straight lines

We answer some questions of Tverberg about separability properties of families of convex sets. In particular, we show that there is a family of infinitely many pairwise disjoint closed disks, no two of which can be separated from two others by a straight line. No such construction exists with equal disks. We also prove that every uncountable family of pairwise disjoint convex sets in the plane has two uncountable subfamilies that can be separated by a straight line.     

有势算子对耦合物理量的分离

本文通过有势算子对耦合物理量的分离推导了耦含热弹性动力学中的两个特殊的最小值原理．     

分离向量,分离泛函与自反性

本文利用严格分离向量和严格分离泛函讨论了算子子空间的自反性及遗传自反性     

Separating two simple polygons by a sequence of translations

LetP andQ be two disjoint simple polygons havingm andn sides, respectively. We present an algorithm which determines whetherQ can be moved by a sequence of translations to a position sufficiently far fromP without colliding withP, and which produces such a motion if it exists. Our algorithm runs in timeO(mn(mn) logm logn) where (k) is the extremely slowly growing inverse Ackermann's function. Since in the worst case (mn) translations may be necessary to separateQ fromP, our algorithm is close to optimal.Work on this paper by the first author has been supported by National Science Foundation Grant No. DMS-8501947. Work on this paper by the second author has been supported by Office of Naval Research Grant No. N00014-82-K-0381, National Science Foundation Grant No. NSP-DCR-83-20085, and by grants from the Digital Equipment Corporation, and the IBM Corporation. Work by the second and third authors has also been supported by a grant from the joint Ramot-Israeli Ministry of Industry Foundation. Part of the work on this paper has been carried out at the Workshop on Movable Separability of Sets at the Bellairs Research Institute of McGill University, Barbados, February 1986.     

Subadditive Separating Maps

Let X and Y denote compact Hausdorff spaces and let K = R (real numbers) or C(complex numbers). C(X) and C(Y) denote the spaces of K-valued continuous functions on X and Y, respectively. A map H : C(X) C(Y) is separating if fg = 0 implies that HfHg = 0. Results about automatic continuity and the form of additive and linear separating maps have been developed in [1], [2], [3], [4], [5], [7], [8], and [10]. In this article similar results are developed for subadditive separating maps. We show (Theorem 5.11) that certain biseparating, subadditive bijections H are automatically continuous.     

Separating principles below

In this paper, we study the Ramsey‐type weak K?nig's Lemma, written , using a technique introduced by Lerman, Solomon, and the second author. This technique uses iterated forcing to construct an ω‐model satisfying one principle T₁ but not another T₂. The technique often allows one to translate a “one step” construction (building an instance of T₂ along with a collection of solutions to each computable instance of T₁) into an ω‐model separation (building a computable instance of T₂ together with a Turing ideal where T₁ holds but this instance has no solution). We illustrate this translation by separating from (reproving a result of Ambos‐Spies, Kjos‐Hanssen, Lempp, and Slaman), and then apply this technique to separate from (which has been shown separately by Bienvenu, Patey, and Shafer).     

Separating sets by Darboux Baire 1 and approximately continuous functions

We characterize sets A₀, A₁ for which there is a DB1 function f with [f = 0] = A₀ and [f = 1] = A₁. This characterization is a conjunction of necessary conditions for Darboux and for Baire 1 functions. We also characterize sets A^?, A⁺ for which there is a DB₁ function with [f < 0] = A^? and [f > 0] = A⁺. The same characterzations are provided for approximately continuous functions.     

Separating Sets as Envelopes

The set which forms the boundary of the regions of interferencewhen a family of curves is superposed on a parallel gratingis related to the envelopes of two ‘interference sets’in two fixed directions.     

Separating vectors for operators

It is an open problem whether every one-dimensional extension of a triangular operator admits a separating vector. We prove that the answer is positive for many triangular Hilbert space operators, and in particular, for strictly triangular operators. This is revealing, because two-dimensional extensions of such operators can fail to have separating vectors.

     

Separating subadditive euclidean functionals

The classical Beardwood‐Halton‐Hammersly theorem (1959) asserts the existence of an asymptotic formula of the form for the minimum length of a Traveling Salesperson Tour throuh n random points in the unit square, and in the decades since it was proved, the existence of such formulas has been shown for other such Euclidean functionals on random points in the unit square as well. Despite more than 50 years of attention, however, it remained unknown whether the minimum length TSP through n random points in was asymptotically distinct from its natural lower bounds, such as the minimum length spanning tree, the minimum length 2‐factor, or, as raised by Goemans and Bertsimas, from its linear programming relaxation. We prove that the TSP on random points in Euclidean space is indeed asymptotically distinct from these and other natural lower bounds, and show that this separation implies that branch‐and‐bound algorithms based on these natural lower bounds must take nearly exponential () time to solve the TSP to optimality, even in average case. This is the first average‐case superpolynomial lower bound for these branch‐and‐bound algorithms (a lower bound as strong as was not even been known in worst‐case analysis). © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 375–403, 2017     

Morita Equivalent Blocks in Non-Normal Subgroups and p-Radical Blocks in Finite Groups

Let O be a complete discrete valuation ring with unique maximalideal J(O), let K be its quotient field of characteristic 0,and let k be its residue field O/J(O) of prime characteristicp. We fix a finite group G, and we assume that K is big enoughfor G, that is, K contains all the |G|-th roots of unity, where|G| is the order of G. In particular, K and k are both splittingfields for all subgroups of G. Suppose that H is an arbitrarysubgroup of G. Consider blocks (block ideals) A and B of thegroup algebras RG and RH, respectively, where R{O, k}. We considerthe following question: when are A and B Morita equivalent?Actually, we deal with ‘naturally Morita equivalent blocksA and B’, which means that A is isomorphic to a full matrixalgebra of B, as studied by B. Külshammer. However, Külshammerassumes that H is normal in G, and we do not make this assumption,so we get generalisations of the results of Külshammer.Moreover, in the case H is normal in G, we get the same resultsas Külshammer; however, he uses the results of E. C. Dade,and we do not.