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In this paper we introduce and study a two-parameter family of integral operators on the Fock space F 2 ( C ) . We determine exactly when these operators are bounded and when they are unitary. We show that, under the Bargmann transform, these operators include the classical linear canonical transforms as special cases. As an application, we obtain a new unitary projective representation for the special linear group S L ( 2 , R ) on the Fock space.

Let be a set of primes. A finite group  is a  -group if all prime divisors of the order of  belong to  . Following Wielandt, the -Sylow theorem holds for if all maximal -subgroups of are conjugate; if the -Sylow theorem holds for every subgroup of  then the strong -Sylow theorem holds for  . The strong -Sylow theorem is known to hold for  if and only if it holds for every nonabelian composition factor of  . In 1979, Wielandt asked which finite simple nonabelian groups obey the strong -Sylow theorem. By now the answer is known for sporadic and alternating groups. We give some arithmetic criterion for the validity of the strong -Sylow theorem for the groups  .

We provide an explicit construction of finite 4-regular graphs (Γk)k∈N with girthΓk→∞ as k→∞ and diamΓkgirthΓk⩽D for some D>0 and all k∈N. For each fixed dimension n⩾2, we find a pair of matrices in SLn(Z) such that (i) they generate a free subgroup, (ii) their reductions modp generate SLn(Fp) for all sufficiently large primes p, (iii) the corresponding Cayley graphs of SLn(Fp) have girth at least cnlogp for some cn>0. Relying on growth results (with no use of expansion properties of the involved graphs), we observe that the diameter of those Cayley graphs is at most O(logp). This gives infinite sequences of finite 4-regular Cayley graphs of SLn(Fp) as p→∞ with large girth and bounded diameter-by-girth ratio. These are the first explicit examples in all dimensions n⩾2 (all prior examples were in n=2). Moreover, they happen to be expanders. Together with Margulis’ and Lubotzky–Phillips–Sarnak’s classical constructions, these new graphs are the only known explicit logarithmic girth Cayley graph expanders.

In this work we find solutions of the ( n + 2 )-dimensional Einstein Field Equations (EFE) with n commuting Killing vectors in vacuum. In the presence of n Killing vectors, the EFE can be separated into blocks of equations. The main part can be summarized in the chiral equation ( α g , z ¯ g - 1 ) , z + ( α g , z g - 1 ) , z ¯ = 0 with g ∈ S L ( n , R ) . The other block reduces to the differential equation ( ln f α 1 - 1 / n ) , z = 1 / 2 α tr ( g , z g - 1 ) 2 and its complex conjugate. We use the ansatz g = g ( ξ ) , where ξ satisfies a generalized Laplace equation, so the chiral equation reduces to a matrix equation that can be solved using algebraic methods, turning the problem of obtaining exact solutions for these complicated differential equations into an algebraic problem. The different EFE solutions can be chosen with desired physical properties in a simple way.

A perfect code in a graph Γ is a subset C of V ( Γ ) such that no two vertices in C are adjacent and every vertex in V ( Γ ) \\ C is adjacent to exactly one vertex in C . Let G be a finite group and C a subset of G . Then C is said to be a perfect code of G if there exists a Cayley graph of G admiting C as a perfect code. It is proved that a subgroup H of G is a perfect code of G if and only if a Sylow 2-subgroup of H is a perfect code of G . This result provides a way to simplify the study of subgroup perfect codes of general groups to the study of subgroup perfect codes of 2-groups. As an application, a criterion for determining subgroup perfect codes of projective special linear groups PSL ( 2 , q ) is given.

To a finite dimensional representation of a complex Lie group G , an associative algebra of adjoint covariant polynomial maps from the direct sum of m copies of the Lie algebra g of G into an algebra of complex matrices is associated. When the tangent representation of the given representation is irreducible, the center of this algebra of concomitants can be identified with the algebra of adjoint invariant polynomial functions on m -tuples of elements of g . For irreducible finite dimensional representations of SL 2 ( ℂ ) minimal generating systems of the corresponding algebras of concomitants are determined, both as an algebra and as a module over its center.

The question is still open as to whether there exist infinitely many Fermat primes or infinitely many composite Fermat numbers. The same question concerning Mersenne numbers is also unanswered. Extending some recent results of Megrelishvili and the author, we characterize the Fermat primes and the Mersenne primes in terms of the topological minimality of some matrix groups. This is achieved by showing, among other things, that if F is a subfield of a local field of characteristic ≠2, then the special upper triangular group ST+(n,F) is minimal precisely when the special linear group SL(n,F) is. We provide criteria for the minimality (and total minimality) of SL(n,F) and ST+(n,F), where F is a subfield of C. Let Fπ and Fc be the set of Fermat primes and the set of composite Fermat numbers, respectively. As our main result, we prove that the following conditions are equivalent for A∈Fπ,Fc: A is finite; ∏Fn∈ASL(Fn−1,Q(i)) is minimal, where Q(i) is the Gaussian rational field; and ∏Fn∈AST+(Fn−1,Q(i)) is minimal. Similarly, denote by Mπ and Mc the set of Mersenne primes and the set of composite Mersenne numbers, respectively, and let B∈Mπ,Mc. Then the following conditions are equivalent: B is finite; ∏Mp∈BSL(Mp+1,Q(i)) is minimal; and ∏Mp∈BST+(Mp+1,Q(i)) is minimal.

We classify Weingarten conoids in the real special linear group SL(2,R). In particular, there is no linear Weingarten nontrivial conoids in SL(2,R). We also prove that the only conoids in SL(2,R) with constant Gaussian curvature are the flat ones. Finally, we show that any surface that is invariant under left translations of the subgroup N is a Weingarten surface.

Given a group G and a subgroup H ≤ G , a set F ⊂ G is called H -intersecting if for any g , g ′ ∈ F , there exists x H ∈ G / H such that g x H = g ′ x H . The intersection density of the action of G on G / H by (left) multiplication is the rational number ρ ( G , H ) , equal to the maximum ratio | F | | H | , where F ⊂ G runs through all H -intersecting sets of G . The intersection spectrum of the group G is then defined to be the set σ ( G ) : = ρ ( G , H ) : H ≤ G . It was shown by Bardestani and Mallahi-Karai (J Algebraic Combin, 42(1):111–128, 2015) that if σ ( G ) = { 1 } , then G is necessarily solvable. The natural question that arises is, therefore, which rational numbers larger than 1 belong to σ ( G ) , whenever G is non-solvable. In this paper, we study the intersection spectrum of the linear group PSL 2 ( q ) . It is shown that 2 ∈ σ PSL 2 ( q ) , for any prime power q ≡ 3 ( mod 4 ) . Moreover, when q ≡ 1 ( mod 4 ) , it is proved that ρ ( PSL 2 ( q ) , H ) = 1 , for any odd index subgroup H (containing F q ) of the Borel subgroup (isomorphic to F q ⋊ Z q - 1 2 ) consisting of all upper triangular matrices.

Abstract We prove that every element of the special linear group can be represented as the product of at most six block unitriangular matrices, and that there exist matrices for which six products are necessary, independent of indexing. We present an analogous result for the general linear group. These results serve as general statements regarding the representational power of alternating linear updates. The factorizations and lower bounds of this work immediately imply tight estimates on the expressive power of linear affine coupling blocks in machine learning.