Small modules over quantum complete intersections in two variables

We describe the left regular module of a quantum complete intersection [Formula: see text] by the property that it is the unique finite-dimensional indecomposable left [Formula: see text]-module of Loewy length [Formula: see text] Using a reduction to [Formula: see text]-modules, we classify the [Formula: see text]-dimensional indecomposable left modules over quantum complete intersection [Formula: see text] in two variables, by explicitly giving their diagram presentations. Together with the existed work on indecomposable [Formula: see text]-modules of dimension [Formula: see text], we then know all the indecomposable [Formula: see text]-modules of dimension [Formula: see text].

FINE DELIGNE–LUSZTIG VARIETIES AND ARITHMETIC FUNDAMENTAL LEMMAS

We prove a character formula for some closed fine Deligne–Lusztig varieties. We apply it to compute fixed points for fine Deligne–Lusztig varieties arising from the basic loci of Shimura varieties of Coxeter type. As an application, we prove an arithmetic intersection formula for certain diagonal cycles on unitary and GSpin Rapoport–Zink spaces arising from the arithmetic Gan–Gross–Prasad conjectures. In particular, we prove the arithmetic fundamental lemma in the minuscule case, without assumptions on the residual characteristic.

Operations on Unambiguous Finite Automata

A nondeterministic finite automaton is unambiguous if it has at most one accepting computation on every input string. We investigate the state complexity of basic regular operations on languages represented by unambiguous finite automata. We get tight upper bounds for reversal ([Formula: see text]), intersection ([Formula: see text]), left and right quotients ([Formula: see text]), positive closure ([Formula: see text]), star ([Formula: see text]), shuffle ([Formula: see text]), and concatenation ([Formula: see text]). To prove tightness, we use a binary alphabet for intersection and left and right quotients, a ternary alphabet for star and positive closure, a five-letter alphabet for shuffle, and a seven-letter alphabet for concatenation. For complementation, we reduce the trivial upper bound [Formula: see text] to [Formula: see text]. We also get some partial results for union and square.

On the existence of E0-semigroups — the multiparameter case

Let [Formula: see text] be a closed convex cone. Assume that [Formula: see text] is pointed, i.e. the intersection [Formula: see text] and [Formula: see text] is spanning, i.e. [Formula: see text]. Denote the interior of [Formula: see text] by [Formula: see text]. Let [Formula: see text] be a product system over [Formula: see text]. We show that there exists an infinite-dimensional separable Hilbert space [Formula: see text] and a semigroup [Formula: see text] of unital normal ∗-endomorphisms of [Formula: see text] such that [Formula: see text] is isomorphic to the product system associated to [Formula: see text].

INTERSECTIONS OF TRANSLATION OF A CLASS OF SIERPINSKI CARPETS

For [Formula: see text], a middle-[Formula: see text] Sierpinski carpet [Formula: see text] is defined as the self-similar set generated by the iterated function system (IFS) [Formula: see text], where [Formula: see text] is defined by [Formula: see text] Here, [Formula: see text]. In this paper, for [Formula: see text], we investigated the equivalent characterizations of the intersection [Formula: see text] being a generalized Moran set. Furthermore, under some conditions, we show that [Formula: see text] can be represented as a graph-directed set satisfying the open set condition (OSC), and then the Hausdorff dimension can be explicitly calculated.

On the intersection of tame subgroups in groups acting on trees

Let [Formula: see text] be a group acting on a tree [Formula: see text] with finite edge stabilizers of bounded order. We provide, in some very interesting cases, upper bounds for the complexity of the intersection [Formula: see text] of two tame subgroups [Formula: see text] and [Formula: see text] of [Formula: see text] in terms of the complexities of [Formula: see text] and [Formula: see text]. In particular, we obtain bounds for the Kurosh rank [Formula: see text] of the intersection in terms of Kurosh ranks [Formula: see text] and [Formula: see text], in the case, where [Formula: see text] and [Formula: see text] act freely on the edges of [Formula: see text].

APPLICATIONS OF AN INTERSECTION FORMULA TO DUAL CONES

We give a succinct proof of a duality theorem obtained by Révész [‘Some trigonometric extremal problems and duality’, J. Aust. Math. Soc. Ser. A 50 (1991), 384–390] which concerns extremal quantities related to trigonometric polynomials. The key tool of our new proof is an intersection formula on dual cones in real Banach spaces. We show another application of this intersection formula which is related to integral estimates of nonnegative positive-definite functions.

INFLATING BALLS IS NP-HARD

A collection [Formula: see text] of balls in ℝd is δ-inflatable if it is isometric to the intersection [Formula: see text] of some d-dimensional affine subspace E with a collection [Formula: see text] of (d + δ)-dimensional balls that are disjoint and have equal radius. We give a quadratic-time algorithm to recognize 1-inflatable collections of balls in any fixed dimension, and show that recognizing δ-inflatable collections of d-dimensional balls is NP-hard for δ ≥ 2 and d ≥ 3 if the balls' centers and radii are given by numbers of the form [Formula: see text] where a, …, e are integers.