论文标题
合理函数的合并式con症，几乎严格地是单峰序列和理想玻色纤维气体的相变
Merged-log-concavity of rational functions, almost strictly unimodal sequences, and phase transitions of ideal boson-fermion gases
论文作者
论文摘要
我们通过具有正整数系数的多项式的真实函数的实际值的单峰序列获得了一些新结果。因此，我们介绍了合理函数的合并现象con症的概念。粗略地说，这一概念扩展了斯坦利的$ q $ log-concavity of tolyenmials。 我们通过$ Q $ binmial系数，Hadamard产品和卷积构建明确合并的合理功能，以扩展Cauchy-Binet公式。然后，我们通过Young图获得了合理函数的单峰序列。此外，我们考虑了单峰序列通过临界点的变化，这些临界点在几乎严格的单峰序列之间严格增加，严格减少，严格减少和山形序列。同样，关键点是在合适的环境中多项式的零。 上面的研究扩展了$（\ pm t; q）的$ t $ power系列_ {\ infty}^{\ mp 1} $在某种程度上通过具有正整数系数的多项式和单峰序列的变化。然后，我们将量子差异（$ q $ - 指数）的黄金比例作为关键点。此外，我们考虑了ETA产品，通用的Narayana数量以及加权$ Q $ - 多数系数，我们引入了。 在统计力学中，我们讨论了具有或没有Casimir能量的某些理想玻色子 - 弗利米亚气体的大规范分区功能（Ramanujan求和）。合并的concavity通过包括黄金比率在内的金属比的临界点对Helmholtz自由能进行了相变。特别是，随着温度的升高，相变的相变表示来自零粒子真空吸尘器的非零粒子真空吸尘器。
We obtain some new results on the unimodal sequences of the real values of rational functions by polynomials with positive integer coefficients. Thus, we introduce the notion of merged-log-concavity of rational functions. Roughly speaking, the notion extends Stanley's $q$-log-concavity of polynomials. We construct explicit merged-log-concave rational functions by $q$-binomial coefficients, Hadamard products, and convolutions, extending the Cauchy-Binet formula. Then, we obtain the unimodal sequences of rational functions by Young diagrams. Moreover, we consider the variation of unimodal sequences by critical points that separate strictly increasing, strictly decreasing, and hill-shape sequences among almost strictly unimodal sequences. Also, the critical points are zeros of polynomials in a suitable setting. The study above extends the $t$-power series of $(\pm t;q)_{\infty}^{\mp 1}$ to some extent by polynomials with positive integer coefficients and the variation of unimodal sequences. We then obtain the golden ratio of quantum dilogarithms ($q$-exponentials) as a critical point. Additionally, we consider eta products, generalized Narayana numbers, and weighted $q$-multinomial coefficients, which we introduce. In statistical mechanics, we discuss the grand canonical partition functions of some ideal boson-fermion gases with or without Casimir energies (Ramanujan summation). The merged-log-concavity gives phase transitions on Helmholtz free energies by critical points of the metallic ratios including the golden ratio. In particular, the phase transitions implies non-zero particle vacua from zero particle vacua as the temperature rises.
