A q-analogue of Catalan Hankel determinants

In this paper we shall survey the various methods of evaluating Hankel determinants and as an illustration we evaluate some Hankel determinants of a q-analogue of Catalan numbers. Here we consider $\frac{(aq;q)_{n}}{(abq^{2};q)_{n}}$ as a q-analogue of Catalan numbers $C_{n}=\frac1{n+1}\binom{2n}{n}$, which is known as the moments of the little q-Jacobi polynomials. We also give several proofs of this q-analogue, in which we use lattice paths, the orthogonal polynomials, or the basic hypergeometric series. We also consider a q-analogue of Schr\"oder Hankel determinants, and give a new proof of Moztkin Hankel determinants using an addition formula for ${}_2F_{1}$.Comment: 17 page

Generalizations of Cauchy's Determinant and Schur's Pfaffian

We present several generalizations of Cauchy's determinant and Schur's Pfaffian by considering matrices whose entries involve some generalized Vandermonde determinants. Special cases of our formulae include previuos formulae due to S.Okada and T. Sundquist. As an application, we give a relation for the Littlewood--Richardson coefficients involving a rectangular partition.Comment: 26 page

Impact of hypoxia on the pathogenesis and therapy resistance in multiple myeloma

Multiple myeloma (MM) is a refractory plasma cell tumor. In myeloma cells, the transcription factor IRF4, the master regulator of plasma cells, is aberrantly upregulated and plays an essential role in oncogenesis. IRF4 forms a positive feedback loop with MYC, leading to additional tumorigenic properties. In recent years, molecular targeted therapies have contributed to a significant improvement in the prognosis of MM. Nevertheless, almost all patients experience disease progression, which is thought to be a result of treatment resistance induced by various elements of the bone marrow microenvironment. Among these, the hypoxic response, one of the key processes for cellular homeostasis, induces hypoxia-adapted traits such as undifferentiation, altered metabolism, and dissemination, leading to drug resistance. These inductions are caused by ectopic gene expression changes mediated by the activation of hypoxia-inducible factors (HIFs). By contrast, the expression levels of IRF4 and MYC are markedly reduced by hypoxic stress. Notably, an anti-apoptotic capability is usually acquired under both normoxic and hypoxic conditions, but the mechanism is distinct. This fact strongly suggests that myeloma cells may survive by switching their dependent regulatory factors from IRF4 and MYC (normoxic bone marrow region) to HIF (hypoxic bone marrow microenvironment). Therefore, to achieve deep remission, combination therapeutic agents, which are complementarily effective against both IRF4-MYC-dominant and HIF-dominated fractions, may become an important therapeutic strategy for MM