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  4. Abstract How many chess rooks or queens does it take to guard all squares of a given polyomino, the union of square tiles from a square grid? This question is a version of the art gallery problem in which the guards can “see” whichever squares the rook or queen attacks. We show that $$\lfloor {\frac{n}{2}} \rfloor $$ ⌊ n 2 ⌋ rooks or $$\lfloor {\frac{n}{3}} \rfloor $$ ⌊ n 3 ⌋ queens are sufficient and sometimes necessary to guard a polyomino with n tiles. We then prove that finding the minimum number of rooks or queens needed to guard amore »polyomino is NP-hard. These results also apply to d -dimensional rooks and queens on d -dimensional polycubes. Finally, we use bipartite matching theorems to describe sets of non-attacking rooks on polyominoes.« less
  5. We give bijective proofs of Monk’s rule for Schubert and double Schubert polynomials computed with bumpless pipe dreams. In particular, they specialize to bijective proofs of transition and cotransition formulas of Schubert and double Schubert polynomials, which can be used to establish bijections with ordinary pipe dreams.
  6. Abstract We construct a new family of lattice packings for superballs in three dimensions (unit balls for the l 3 p $\begin{array}{}\displaystylel^p_3\end{array}$ norm) with p ∈ (1, 1.58]. We conjecture that the family also exists for p ∈ (1.58, log 2 3 = 1.5849625…]. Like in the densest lattice packing of regular octahedra, each superball in our family of lattice packings has 14 neighbors.
  7. Abstract We report on our results concerning the distribution of the geometric Picard ranks of K 3 surfaces under reduction modulo various primes. In the situation that $${\mathop {{\mathrm{rk}}}\nolimits \mathop {{\mathrm{Pic}}}\nolimits S_{{\overline{K}}}}$$ rk Pic S K ¯ is even, we introduce a quadratic character, called the jump character, such that $${\mathop {{\mathrm{rk}}}\nolimits \mathop {{\mathrm{Pic}}}\nolimits S_{{\overline{{\mathbb {F}}}}_{\!{{\mathfrak {p}}}}} > \mathop {{\mathrm{rk}}}\nolimits \mathop {{\mathrm{Pic}}}\nolimits S_{{\overline{K}}}}$$ rk Pic S F ¯ p > rk Pic S K ¯ for all good primes at which the character evaluates to $$(-1)$$ ( - 1 ) .