国家预印本平台

We study the distortion of the lower dimension and lower spectrum of sets under quasiconformal mappings. As an application, we verify weighted Hardy--Sobolev inequalities in Euclidean domains arising as quasiconformal images of domains with sufficiently regular boundaries. The regularity in question is phrased in terms of estimates for either the Assouad or the lower dimension of the complement. We also provide dilatation-dependent classification results for certain planar open sets.

We study the asymptotic behaviour of the mean square of long truncations of the Dirichlet series for $\bigl(ζ'/ζ\bigr)^{k}$ on the critical line; we establish the connection between this problem and the short-interval variance of the convoluted von Mangoldt function $Λ^{*k}$ in short intervals and in arithmetic progressions. Under a shifted moments conjecture and the Riemann Hypothesis (RH), we prove that the short-interval variance leading coefficient is the derivative of the mean-square leading coefficient. A unitary trace-sum average governs the variance of sums of $Λ^{*k}$ over monic polynomials in $\mathbb F_q[T]$; we denote its leading coefficient by $δ_k(c)$, where the total trace degree and matrix size grow with a fixed ratio $c$. Under a weaker shifted moments conjecture and RH, we recover this $δ_k(c)$ as the leading coefficient of the variance of short-interval sums of $Λ^{*k}$ over the integers. It is the counterpart of $γ_k(c)$, the piecewise polynomial from the variance of divisor sums, with the Dirichlet coefficients of $ζ(s)^k$ replaced by those of $\bigl(-ζ'/ζ\bigr)^k$.

We deal with the following two pick-up sticks problems: given an integer n greater than or equal to 3, we choose n sticks with independent random lengths from an exponential distribution or Pareto type I distribution, and we are interested in the probability that no three of them can be the sides of a triangle. In case of exponential distribution, this problem has been solved by other authors: the probability in question is equal to $n!/\big((F_3-1)(F_4-1)\cdots(F_{n+2}-1)\big)$, where $(F_i)_{i=1}^\infty$ is the sequence of Fibonacci numbers. We give a new, self-contained and detailed proof for this formula, which uses a different method compared to the earlier ones. In case of Pareto type I distribution, we derive new formulae for the corresponding probability.

Let $\La_1$ and $\La_2$ be finite-dimensional $\ta_2$-finite complex algebras of global dimension at most two. We prove that, if they are cluster equivalent, then they admit $\bbZ$-gradings for which the bounded derived categories of graded modules are triangle equivalent. The equivalence is not required to commute with the internal grading shifts. The proof uses common multigraded dg models, relative gradings on two-term silting objects, and covering quotients. We also consider tilting lifts of cluster-tilting objects with endomorphism algebras of global dimension at most two. In acyclic cluster type, the existence of such a lift for every cluster-tilting object implies piecewise heredity. We construct a counterexample without this assumption using a surface algebra associated with a one-holed torus.

In this note, we study the regularity of solutions to the geodesic equation in the space of plurisubharmonic functions in a general Reinhardt domain. Our findings resolve a problem left open from our previous note [AD26].