On 2-(45, 12, 3) designs

Under the assumption that the incidence matrix of a 2-(45, 12, 3) design has a certain block structure, we determine completely the number of nonisomorphic designs involved. We discover 1136 such designs with trivial automorphism group. In addition we analyze all 2-(45, 12, 3) designs having an automorphism of order 5 or 11. Altogether, the total number of nonisomorphic 2-(45, 12, 3) designs found in 3752. Many of these designs are self-dual and each of these self-dual designs possess a polarity. Some have polarities with no absolute points, giving rise to strongly regular (45, 12, 3, 3) graphs. In total we discovered 58 pairwise nonisomorphic strongly regular graphs, one of which has a trivial automorphism group. Further, we analyzed completely all the designs for subdesigns with parameters 2-(12, 4, 3), 2-(9, 3, 3), and 2-(5, 4, 3). In the first case, the number of 2-(12, 4, 3) subdesigns that a design possessed, if non-zero, turned out to be a multiple of 3, whereas 2-(9, 3, 3) subdesigns were so abundant it was more unusual to find a design without them. Finally, in the case of 2-(5, 4, 3) subdesigns there is a design, unique amongst the ones discovered, that has precisely 9 such subdesigns and these form a partition of the point set of the design. This design has a transitive group of automorphisms of order 360. © 1996 John Wiley & Sons, Inc.     

Classification of Hadamard matrices of order 44 with automorphisms of order 7

The Hadamard matrices of order 44 possessing automorphisms of order 7 are classified. The number of their equivalence classes is 384. The order of their full automorphism group is calculated. These Hadamard matrices yield 1683 nonisomorphic 3-(44,22,10) designs, 57932 nonisomorphic 2-(43,21,10) designs, and two inequivalent extremal binary self-dual doubly even codes of length 88 (one of them being new).     

On representing block designs in terms of their automorphisms

Block designs are analyzed in terms of the structure imposed upon them by their automorphisms. An extension of the notion of a difference set is used to describe necessary and sufficient conditions for the existence of a given automorphism acting on the design. In addition, it is shown that the possible point and block orbit configurations relative to an automorphism acting on a design are rather limited. The development is carried out with a view toward finding unknown designs and studying the automorphism groups of known designs.     

Hadamard matrices of order 28 with automorphisms of order 7

The only primes which can divide the order of the automorphism group of a Hadamard matrix of order 28 are 13, 7, 3, and 2, and there are precisely four inequivalent matrices with automorphisms of order 13 (Tonchev, J. Combin. Theory Ser. A35 (1983), 43–57). In this paper we show that there are exactly twelve inequivalent Hadamard matrices of order 28 with automorphisms of order 7. In particular, there are precisely seven matrices with transitive automorphism groups.     

On automorphism groups of connected Lie groups

We prove that ifG is a connected Lie group with no compact central subgroup of positive dimension then the automorphism group ofG is an almost algebraic subgroup of , where is the Lie algebra ofG. We also give another proof of a theorem of D. Wigner, on the connected component of the identity in the automorphism group of a general connected Lie group being almost algebraic, and strengthen a result of M.Goto on the subgroup consisting of all automorphisms fixing a given central element.     

Similarity automorphism invariance of some P-properties of linear transformations on Euclidean Jordan algebras

Recently, Gowda and Sznajder [Gowda, M.S., Sznajder, R.: Automorphism invariance of P- and GUS-properties of linear transformations on Euclidean Jordan algebras. Math. Oper. Res. 31, 109–123 (2006)] have introduced and studied automorphism invariance of some P-properties for linear transformations. This paper deals with this automorphism invariance of some other complementarity properties, such as \(\hbox {E}_0,\,\hbox {P}_0\) , S, Z-properties. Particularly, we answer Gowda and Sznajder in positive that order P-property is algebra automorphism invariant in simple Jordan algebras. By replacing transposition with the invertibility in the concept of automorphism invariance, we propose a notion of similarity automorphism invariance. Most complementarity properties of linear transformations are also shown to be similarity invariant under algebra automorphisms and cone automorphisms.     

Automorphisms of groups of orderp 5

We study automorphisms of groups of orderp ⁵ (p is an odd prime number). Groups without any automorphism of order 2 and groups with group automorphisms of orderp ⁶ are found.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 4, pp. 562–565, April, 1995.     

Semigroups of{\cal I} - density continuous functionscontinuous functions

This paper is concerned with the classes of continuous functions and continuous functions as semigroups with composition as the operation. We also analyze some of their subsemigroups. It is shown that the groups of automorphisms of these semigroups and several of their subsemigroups have the inner automorphism property. Communicated by M. W. Mislove     

Fixed points of automorphisms of real algebraic curves

We bound the number of fixed points of an automorphism of a real curve in terms of the genus and the number of connected components of the real part of the curve. Using this bound, we derive some consequences concerning the maximum order of an automorphism and the maximum order of an abelian group of automorphisms of a real curve. We also bound the full group of automorphisms of a real hyperelliptic curve. Work supported by the European Community’s Human Potential Programme under contract HPRN-CT-2001-00271, RAAG.     

A Matrix Poincaré formula for holomorphic automorphisms of quadrics of higher codimension. Real Associative Quadrics

In this paper we describe the holomorphic automorphisms for two infinite series of Hermitian quadrics: quadrics of real co-dimension 2 in ℂⁿ⁺² and a special class of quadrics of co-dimension n in ℂ²ⁿ with large automorphism groups (Real Associative Quadrics). We give explicit formulas of the automorphisms. They are rational maps of degree not exceeding the co-dimension.     

Factor group of the automorphism group of a matroid basis graph with respect to the automorphism group of the matroid

Any automorphism of a matroid induces an automorphism of its basis graph. We try to determine what can be said concerning the automorphisms of the basis graph which are not induced by matroids' automorphisms. In particular, we determine the structure of the factor group of the automorphism group of the basis graph with respect to the automorphism group of the matroid, in the event that this factor group exists.     

Constructing flag-transitive,point-imprimitive designs

We give a construction of a family of designs with a specified point-partition and determine the subgroup of automorphisms leaving invariant the point-partition. We give necessary and sufficient conditions for a design in the family to possess a flag-transitive group of automorphisms preserving the specified point-partition. We give examples of flag-transitive designs in the family, including a new symmetric 2-(1408,336,80) design with automorphism group \(2^{12}:((3\cdot \mathrm {M}_{22}):2)\) and a construction of one of the families of the symplectic designs (the designs \(S^-(n)\)) exhibiting a flag-transitive, point-imprimitive automorphism group.     

The bigraphical t-wise balanced designs of index 1

The t-wise balanced designs of index 1 whose point set are the edges of K_m,n, the complete bipartite graph, and that have the subgroup of all automorphisms that fix the two independent sets of K_m,n as an automorphism group are studied. All such designs are found. © 1994 John Wiley & Sons, Inc.     

Wild automorphisms of varieties with Kodaira dimension 0

An automorphism σ of a projective variety X is said to be wild if σ(Y) ≠ Y for every non-empty subvariety Y \subsetneq X{Y \subsetneq X} . In [1] Z. Reichstein, D. Rogalski, and J.J. Zhang conjectured that if X is an irreducible projective variety admitting a wild automorphism then X is an abelian variety, and proved this conjecture for dim(X) ≤ 2. As a step toward answering this conjecture in higher dimensions we prove a structure theorem for projective varieties of Kodaira dimension 0 admitting wild automorphisms. This essentially reduces the Kodaira dimension 0 case to a study of Calabi-Yau varieties, which we also investigate. In support of this conjecture, we show that there are no wild automorphisms of certain Calabi-Yau varieties.     

A Chebotarev Theorem for finite homogeneous extensions of shifts

We derive a Chebotarev Theorem for finite homogeneous extensions of shifts of finite type. These extensions are of the form :X×G/H→X×G/H where (x,gH)=(σx, α(x)gH), for some finite groupG and subgroupH. Given a σ-closed orbit τ, the periods of the -closed orbits covering τ define a partition of the integer |G/H|. The theorem then gives us an asymptotic formula for the number of closed orbits with respect to the various partitions of the integer |G/H|. We apply our theorem to the case of a finite extension and of an automorphism extension of shifts of finite type. We also give a further application to ‘automorphism extensions’ of hyperbolic toral automorphisms. Financially supported by Universiti Kebangsaan Malaysia     

Polynomial automorphisms and hypercyclic operators on spaces of analytic functions

We consider hypercyclic composition operators on which can be obtained from the translation operator using polynomial automorphisms of . In particular we show that if C _S is a hypercyclic operator for an affine automorphism S on , then for some polynomial automorphism Θ and vectors a and b, where I is the identity operator. Finally, we prove the hypercyclicity of “symmetric translations” on a space of symmetric analytic functions on ℓ₁. Received: 8 June 2006 Revised: 26 September 2006     

Automorphisms of Sylow <Emphasis Type="Italic">p</Emphasis>-subgroups of Chevalley groups over <Emphasis Type="Italic">p</Emphasis>-primary residue rings of integers

In this paper, we prove that any automorphism of a Sylow p-subgroup of the Chevalley group over the ring (where p is prime and m ≥ 1) is a product of graph, inner, diagonal, and hypercentral automorphisms. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 8, pp. 121–158, 2006.     

On computable automorphisms in formal concept analysis

Under study are the automorphism groups of computable formal contexts. We give a general method to transform results on the automorphisms of computable structures into results on the automorphisms of formal contexts. Using this method, we prove that the computable formal contexts and computable structures actually have the same automorphism groups and groups of computable automorphisms. We construct some examples of formal contexts and concept lattices that have nontrivial automorphisms but none of them could be hyperarithmetical in any hyperarithmetical presentation of these structures. We also show that it could be happen that two formal concepts are automorphic but they are not hyperarithmetically automorphic in any hyperarithmetical presentation.