Euler measure as generalized cardinality

The simultaneous multiplication formulas for Apostol-Bernoulli polynomials and generalized Frobenius-Euler polynomials

We prove the simultaneous multiplication formulas for Apostol-Bernoulli polynomials and generalized Frobenius-Euler polynomials. These formulas contain Dedekind-Rademacher sums, Apostol-Dedekind sums and Fourier-Dedekind sums.

Sparse recovery guarantees for block orthogonal binary matrices constructed via Generalized Euler Squares

In recent times, the construction of deterministic matrices has gained popularity as an alternative of random matrices as they provide guarantees for recovery of sparse signals. In particular, the construction of binary matrices has attained significance due to their potential for hardware-friendly implementation and appealing applications. Our present work aims at constructing incoherent binary matrices consisting of orthogonal blocks with small block coherence. We show that the binary matrices constructed from Euler squares exhibit block orthogonality and possess low block coherence. With a …

Euler characteristic of the truncated order complex of generalized noncrossing partitions

The purpose of this note is to complete the study, begun in the first author's PhD thesis, of the topology of the poset of generalized noncrossing partitions associated to real reflection groups. In particular, we calculate the Euler characteristic of this poset with the maximal and minimal elements deleted. As we show, the result on the Euler characteristic extends to generalized noncrossing partitions associated to well-generated complex reflection groups.

图的广义锚定构型空间的同调与欧拉示性数

Homology and Euler characteristic of generalized anchored configuration spaces of graphs

本文研究图上 n 个带标记点的广义锚定构型空间，给出非树连通图情形的非交错欧拉示性数公式，并完全确定圆图上相应空间的同调群。

Expansions of generalized Euler's constants into the series of polynomials in $π^{-2}$ and into the formal enveloping series with rational coefficients only

In this work, two new series expansions for generalized Euler's constants (Stieltjes constants) $γ_m$ are obtained. The first expansion involves Stirling numbers of the first kind, contains polynomials in $π^{-2}$ with rational coefficients and converges slightly better than Euler's series $\sum n^{-2}$. The second expansion is a semi-convergent series with rational coefficients only. This expansion is particularly simple and involves Bernoulli numbers with a non-linear combination of generalized harmonic numbers. It also permits to derive an interesting estimation for generalized Euler's cons…

Exact counting of Euler Tours for generalized series-parallel graphs

We give a simple polynomial-time algorithm to exactly count the number of Euler Tours (ETs) of any Eulerian generalized series-parallel graph, and show how to adapt this algorithm to exactly sample a random ET of the given generalized series-parallel graph. Note that the class of generalized seriesparallel graphs includes all outerplanar graphs. We can perform the counting in time $O(mΔ^3)$, where $Δ$ is the maximum degree of the graph with $m$ edges. We use $O(mΔ^2 \log Δ)$ bits to store intermediate values during our computations. To date, these are the first known polynomial-time algorithms…

广义欧拉数与有序集合分划

Generalized Euler numbers and ordered set partitions

本研究针对广义欧拉数研究以代数方法为主的现状，提出将其表示为有序分划带符号和的组合模型，用符号反转对合、偏序集Möbius反演证明了递推、整性、同余式等新旧结果。

Generalized Sidon sets

We give asymptotic sharp estimates for the cardinality of a set of residue classes with the property that the representation function is bounded by a prescribed number. We then use this to obtain an analogous result for sets of integers, answering an old question of Simon Sidon.