A new construction for Williamson-type matrices

It is shown that ifq is a prime power then there are Williamson-type matrices of order

1. 1/2q ²(q + 1) whenq ≡ 1 (mod 4).
2. 1/4q ²(q + 1) whenq ≡ 3 (mod 4) and there are Williamson-type matrices of order 1/4(q + 1).

This gives Williamson-type matrices for the new orders 363, 1183, 1805, 2601, 3174, 5103. The construction can be combined with known results on orthogonal designs to give an Hadamard matrix of the new order 33396 = 4 ? 8349.     

Die Endlichen Polardreiseitgeometrien

LetG be a finite group which is generated by a subsetS of involutions satisfying the theorem of the three reflections: Ifa,b,x,y,z ∈ S, ab ≠ 1 and ifabx,aby,abz are involutions, thenxyz ∈ S. Assume thatS contains three elements which generate a four-group. IfS contains four elements of which no three have a product of order two, then one of the following occurs.

1. G?PGL(2,n), n≡1 (mod 2).
2. G?PSL(2,n), n≡1 (mod 2) and n≥5.
3. G?PSU(3,16).
4. G/Z(G)?PSL(2,9) with ¦Z(G)¦=3.

     

On projective planes admitting elations and homologies

We consider projective planes Π of ordern with abelian collineation group Γ of ordern(n?1) which is generated by (A, m)-elations and (B, l)-homologies wherem =AB andA εl. We prove

1. Ifn is even thenn=2^e and the Sylow 2-subgroup of Γ is elementary abelian.
2. Ifn is odd then the Sylow 2-subgroup of Γ is cyclic.
3. Ifn is a prime then Π is Desarguesian.
4. Ifn is not a square thenn is a prime power.

     

Pancyclism in Chvátal-Erdös's graphs

LetG = (X, E) be a simple graph of ordern, of stability numberα and of connectivityk withα ≤ k. The Chvátal-Erdös's theorem [3] proves thatG is hamiltonian. We have investigated under these conditions what can be said about the existence of cycles of lengthl. We have obtained several results:

1. IfG ≠ K _k,k andG ≠ C ₅,G has aC _n?1 .
2. IfG ≠ C ₅, the girth ofG is at most four.
3. Ifα = 2 and ifG ≠ C ₄ orC ₅,G is pancyclic.
4. Ifα = 3 and ifG ≠ K _3,3,G has cycles of any length between four andn.
5. IfG has noC ₃,G has aC _n?2 .

     

Convergence of representation averages and of convolution powers

LetS be a locally compact (σ-compact) group or semi-group, and letT(t) be a continuous representation ofS by contractions in a Banach spaceX. For a regular probability μ onS, we study the convergence of the powers of the μ-averageUx=∫T(t)xdμ(t). Our main results for random walks on a groupG are:

1. if μ is adapted and strictly aperiodic, and generates a recurrent random walk, thenU ⁿ (U-I) converges strongly to 0. In particular, the random walk is completely mixing.
2. If μ×μ is ergodic onG×G, then for every unitary representationT(.) in a Hilbert space,U ⁿ converges strongly to the orthogonal projection on the space of common fixed points. These results are proved for semigroup representations, along with some other results (previously known only for groups) which do not assume ergodicity.
3. If μ is spread-out with supportS, then $\left\| {\mu ^{n + K} - \mu ^n } \right\| \to 0$ if and only if e $ \in \overline { \cup _{j = 0}^\infty S^{ - j} S^{j + K} } .$ .

     

Compactness and the axiom of choice

In the absence of the axiom of choice four versions of compactness (A-, B-, C-, and D-compactness) are investigated. Typical results:

1. C-compact spaces form the epireflective hull in Haus of A-compact completely regular spaces.
2. Equivalent are:
3. the axiom of choice,
4. A-compactness = D-compactness,
5. B-compactness = D-compactness,
6. C-compactness = D-compactness and complete regularity,
7. products of spaces with finite topologies are A-compact,
8. products of A-compact spaces are A-compact,
9. products of D-compact spaces are D-compact,
10. powers X ^k of 2-point discrete spaces are D-compact,
11. finite products of D-compact spaces are D-compact,
12. finite coproducts of D-compact spaces are D-compact,
13. D-compact Hausdorff spaces form an epireflective subcategory of Haus,
14. spaces with finite topologies are D-compact.

1. Equivalent are:
2. the Boolean prime ideal theorem,
3. A-compactness = B-compactness,
4. A-compactness and complete regularity = C-compactness,
5. products of spaces with finite underlying sets are A-compact,
6. products of A-compact Hausdorff spaces are A-compact,
7. powers X ^k of 2-point discrete spaces are A-compact,
8. A-compact Hausdorff spaces form an epireflective subcategory of Haus.

1. Equivalent are:
2. either the axiom of choice holds or every ultrafilter is fixed,
3. products of B-compact spaces are B-compact.

1. Equivalent are:
2. Dedekind-finite sets are finite,
3. every set carries some D-compact Hausdorff topology,
4. every T ₁-space has a T ₁-D-compactification,
5. Alexandroff-compactifications of discrete spaces and D-compact.

     

On a permutablity problem for groups

Letm, n be positive integers. We denote byR(m, n) (respectivelyP(m, n)) the class of all groupsG such that, for everyn subsetsX ₁, X₂, . . .,X _n of sizem ofG there exits a non-identity permutation σ such that $X_1 X_2 ...X_n \cap X_{\sigma (1)} X_{\sigma (2)} ...X_{\sigma (n)} \ne \not 0$ (respectively X₁X₂ . . .X _n = X_σ(1)X{σ(2)} . . . X{gs(n)}). Let G be a non-abelian group. In this paper we prove that

1. G ∈ P(2,3) if and only ifG isomorphic to S₃, whereS _n is the symmetric group onn letters.
2. G ∈ R(2, 2) if and only if¦G¦ ≤ 8.
3. IfG is finite, thenG ∈ R(3, 2) if and only if¦G¦ ≤ 14 orG is isomorphic to one of the following: SmallGroup(16,i), i ∈ {3, 4, 6, 11, 12, 13}, SmallGroup(32,49), SmallGroup(32, 50), where SmallGroup(m, n) is the nth group of orderm in the GAP [13] library.

     

Characterization of joint ergodicity for non-commuting transformations

The study of jointly ergodic transformations, begun in [2] and [1], is continued. The main result is that, ifT ₁,T ₂, …,T _s are arbitrary measure preserving transformations of a probability space (X, ?,μ), then , if and only if the following conditions are satisfied:

1. T ₁×T ₂×…×T _s is ergodic.
2. .

     

Isometric embed-dings between classical Banach spaces,cubature formulas,and spherical designs

If an isometric embeddingl _p ^m →l _q ⁿ with finitep, q>1 exists, thenp=2 andq is an even integer. Under these conditions such an embedding exists if and only ifn?N(m, q) where $$\left( {\begin{array}{*{20}c} {m + q/2 - 1} \\ {m - 1} \\ \end{array} } \right) \leqslant N(m,q) \leqslant \left( {\begin{array}{*{20}c} {m + q - 1} \\ {m - 1} \\ \end{array} } \right).$$ To construct some concrete embeddings, one can use orbits of orthogonal representations of finite groups. This yields:N(2,q)=q/2+1 (by regular (q+2)-gon),N(3, 4)=6 (by icosahedron),N(3, 6)?11 (by octahedron), etc. Another approach is based on relations between embeddings, Euclidean or spherical designs and cubature formulas. This allows us to sharpen the above lower bound forN(m, q) and obtain a series of concrete values, e.g.N(3, 8)=16 andN(7, 4)=28. In the cases (m, n, q)=(3, 6, 10) and (3, 8, 15) some ε-embeddings with ε ～ 0.03 are constructed by the orbit method. The rigidness of spherical designs in Bannai's sense and a similar property for the embeddings are considered, and a conjecture of [7] is proved for any fixed (m, n, q).     

Some results on entire functions of finite lower order

Letf(z) be an entire function of order λ and of finite lower order μ. If the zeros off(z) accumulate in the vicinity of a finite number of rays, then

1. λ is finite;
2. for every arbitrary numberk ₁>1, there existsk ₂>1 such thatT(k ₁ r,f)≤k ₂ T(r,f) for allr≥r ₀. Applying the above results, we prove that iff(z) is extremal for Yang's inequalityp=g/2, then
3. every deficient values off(z) is also its asymptotic value;
4. every asymptotic value off(z) is also its deficient value;
5. λ=μ;
6. $\sum\limits_{a \ne \infty } {\delta (a,f) \leqslant 1 - k(\mu ).} $

     

Some panconnected and pancyclic properties of graphs with a local ore-type condition

Asratian and Khachatrian proved that a connected graphG of order at least 3 is hamiltonian ifd(u) + d(v) ≥ |N(u) ∪ N(v) ∪ N(w)| for any pathuwv withuv ? E(G), whereN(x) is the neighborhood of a vertexx. We prove that a graphG with this condition, which is not complete bipartite, has the following properties:

1. For each pair of verticesx, y with distanced(x, y) ≥ 3 and for each integern, d(x, y) ≤ n ≤ |V(G)| ? 1, there is anx ? y path of lengthn.
2. For each edgee which does not lie on a triangle and for eachn, 4 ≤ n ≤ |V(G)|, there is a cycle of lengthn containinge.
3. Each vertex ofG lies on a cycle of every length from 4 to |V(G)|.

This implies thatG is vertex pancyclic if and only if each vertex ofG lies on a triangle.     

Complex Banach spaces whose duals areL 1-spaces

We prove that for a complex Banach spaceA the following properties are equivalent:

1. A ^* is isometric to anL ₁(μ)-space;
2. every family of 4 balls inA with the weak intersection property has a non-empty intersection;
3. every family of 4 balls inA such that any 3 of them have a non-empty intersection, has a non-empty intersection.

     

On the nonexistence of nontrivial perfecte-codes and tight 2e-designs in hamming schemesH(n,q) withe ≥ 3 andq ≥ 3

It is known that if there exist perfecte-codes or tight 2e-designs in a Hamming schemeH(n, q), then all the zeros of the Lloyd polynomial $$F_e \left( x \right) = \sum\limits_{i = 0}^e {\left( { - q} \right)^i \left( {q - 1} \right)^{e - i} } \left( {\begin{array}{*{20}c} {n - i - 1} \\ {e - i} \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {x - 1} \\ i \\ \end{array} } \right)$$ are integers in the interval [1,n]. With some results of Best, we show that ife ≥ 3,n ≥ e + 1, andq ≥ 3, thenF _e (x) has a nonintegral zero. Therefore there exist no nontrivial perfecte-codes and tight 2e-designs inH(n,q) ife ≥ 3 andq ≥ 3.     

Sulle superficie di uno spazio euclideo an dimensioni aventi un campo tangente concircolare

Letx:M ²→E ⁿ be the immersion of a surfaceM ² in ann-dimensional Euclidean space. Letj and μ be the canonical isomorphism defined by the metricg ofM ² and by the canonical volume element ofM ², respectively. IfM ² carries a concircular tangent vector fieldX. then the following properties are proved:

1. The gaussian curvatureK ofM ² is identically zero.
2. X defines an infinitesimal homothety onM ².
3. The vector field (j ^?1 o μ) (X) is a Killing vector field.

Further ifX is alsoconcurrent (in the sense of K. Yano and B. Y. Chen [3]), then the immersion is degenerate (all the Killing-Lipchitz curvatures associated withx vanish).     

Commutative properties of singularly perturbed operators

Suppose the self-adjoint operatorA in the Hilbert spaceH commutes with the bounded operatorS. Suppose another self-adjoint operatorā is singularly perturbed with respect toA, i.e., it is identical toA on a certain dense set inH. We study the following question: Under what conditions doesā also commute withS? In addition, we consider the case whenS is unbounded and also the case whenS is replaced by a singularly perturbed operator S. As application, we consider the Laplacian inL ₂(R ^q ) that is singularly perturbed by a set of δ functions and commutes with the symmetrization operator inR ^q ,q=2, 3, or with regular representations of arbitrary isometric transformations inR ^q ,q≤3.     

Tilings whose members have finitely many neighbors

Let ? be a tiling of the plane such that each tile of ? meets at most finitely many other tiles. Then exactly one of the following must occur:

1. Uncountably many boundary points of ? belong to no nondegenerate edge of ?, hence ? has uncountably many singular points; or
2. Every boundary point of ? belongs to a nondegenerate edge of ?, moreover, ? has no singular points.

Furthermore, ifS is the set of singular points of ? andW={t:t∈bdry ? andt belongs to no nondegenerate edge of ?}, thenS=clW.     

Homogeneity in infinite permutation groups

It is shown that ifG is a permutation group on an infinite setX, andG is (k?1)-transitive but notk-transitive (wherek ≥ 5), then the following hold:

1. G is not (k + 3)-homogeneous.
2. IfG is (k + 2)-homogeneous, then the group induced byG on ak-subset ofX is the alternating groupA _k .

     

Enumerating rooted simple planar maps

The main purpose of this paper is to find the number of combinatorially distinct rooted simpleplanar maps,i.e.,maps having no loops and no multi-edges,with the edge number given.We haveobtained the following results.1.The number of rooted boundary loopless planar [m,2]-maps.i.e.,maps in which there areno loops on the boundaries of the outer faces,and the edge number is m,the number of edges on theouter face boundaries is 2,is(?)for m≥1.G_0~N=0.2.The number of rooted loopless planar [m,2]-maps is(?)3.The number of rooted simple planar maps with m edges H_m~s satisfies the following recursiveformula:(?)where H_m~(NL) is the number of rooted loopless planar maps with m edges given in [2].4.In addition,γ(i,m),i≥1,are determined by(?)for m≥i.γ(i,j)=0,when i>j.     

A generalization of the katona theorem for crosst-intersecting families

LetX be ann-element set and letA and? be families of subsets ofX. We say thatA and? are crosst-intersecting if |A ∩ B| ≥ t holds for all A ∈A and for allB ∈ ?. Suppose thatA and ? are crosst-intersecting. This paper first proves a crosst-intersecting version of Harper's Theorem:

1. There are two crosst-intersecting Hamming spheresA ₀,? ₀ with centerX such that |A| ≤ |A ₀| and|?| ≤ |? ₀| hold.
2. Suppose thatt ≥ 2 and that the pair of integers (|A) is maximal with respect to direct product ordering among pairs of crosst-intersecting families. Then,A and? are Hamming spheres with centerX.

Using these claims, the following conjecture of Frankl is proven:

1. Ifn + t = 2k ? 1 then |A| |?| ≤ max \(\left\{ {\left( {K_k^n + \left( {_{k - 1}^{n - 1} } \right)} \right)^2 ,K_k^n K_{k - 1}^n } \right\}\) holds, whereK _l ⁿ is defined as \(\left( {_n^n } \right)\left( {_{n - 1}^n } \right) + \cdots + \left( {_l^n } \right).\)
2. Ifn + t = 2k then |A| |? ≤ (K _k ⁿ )² holds.

The extremal configurations are also determined.