it is known that the directed cycle of order $n$ uniquely achieves the minimum spectral radius among all strongly connected digraphs of order $nge 3$. in this paper, among others, we determine the digraphs which achieve the second, the third and the fourth minimum spectral radii respectively among strongly connected digraphs of order $nge 4$.

The multigraded Hilbert scheme parametrizes all homogeneous ideals in a polynomial ring graded by an abelian group with a fixed Hilbert function. We prove that any multigraded Hilbert scheme is smooth and irreducible when the polynomial ring is Z[x, y], which establishes a conjecture of Haiman and Sturmfels.

We study a variety of modes (idempotent algebras with mutually commuting term operations), so called differential modes, having a strongly solvable chain 0 ≤ α ≤ 1 in their congruence lattices. We show an explicit description of subdirectly irreducible algebras in this variety, and use it to compute residual bounds of its subvarieties. It follows from our results that all subvarieties with a fi...

For many years, four–manifold folklore suggested that all simply connected smooth four–manifolds should be connected sums of complex algebraic surfaces, with both their complex and non–complex orientations allowed. The first counterexamples were constructed in 1990 by Gompf and Mrowka [5], and many others followed. Then, Gompf [4] showed that many, and possibly all, these counterexamples arise ...

This paper is a companion to “Sensory Awareness is not a Wide Physical Relation: An Empirical Argument against Externalist Intentionalism” (forthcoming, with a reply by Michael Tye, in Nous). I generalize the argument of that previous paper. I attempt to show that, given the mild form of Internalism argued for in the previous paper, it is not only the case that the awareness relation cannot be ...

We prove that an infinite irreducible Coxeter group cannot be a non-trivial direct product. Let W be a Coxeter group, and write W = W1 × · · · ×Wp ×Wp+1, where W1, . . . ,Wp are infinite irreducible Coxeter groups, and Wp+1 is a finite one. As an application of the main result, we obtain that W1, . . . ,Wp are unique and Wp+1 is unique up to isomorphism. That is, if W = W̃1 × · · · × W̃q × W̃q+1 i...

A permutation π of [n] = {1, 2, . . . , n} is irreducible if π([m]) = [m] for no m ∈ [n], m < n, and it is connected if for no interval I ⊂ [n], 2 ≤ |I| ≤ n − 1, the image π(I) is an interval. We review enumeration of irreducible permutations and their appearances in mathematics. Then we enumerate connected permutations. Asymptotically, there are n!/e2 of them and exactly (for n > 2) their numb...

A finite subgroup of GL(n,C) is involutory if the sum of the dimensions of its irreducible complex representations is given by the number of absolute involutions in the group. A uniform combinatorial model is constructed for all non-exceptional irreducible complex reflection groups which are involutory including, in particular, all infinite families of finite irreducible Coxeter groups.

In this paper, we give a classification of non simply connected seven dimensional Reimannian manifolds of constant positive curvature which admit irreducible cohomogeneity-one actions. We characterize the acting groups and describe the orbits. The first and second homo-topy groups of the orbits have been presented as well.