Boundedness of planar jump discontinuities for homogeneous hyperbolic systems

Suppose that [Formula: see text] is a homogeneous constant coefficient strongly hyperbolic partial differential operator on [Formula: see text] and that [Formula: see text] is a characteristic hyperplane. Suppose that in a conic neighborhood of the conormal variety of [Formula: see text], the characteristic variety of [Formula: see text] is the graph of a real analytic function [Formula: see text] with [Formula: see text] identically equal to zero or the maximal possible value [Formula: see text]. Suppose that the source function [Formula: see text] is compactly supported in [Formula: see text] and piecewise smooth with singularities only on [Formula: see text]. Then the solution of [Formula: see text] with [Formula: see text] for [Formula: see text] is uniformly bounded on [Formula: see text]. Typically when [Formula: see text] on the conormal variety, the sup norm of the jump in the gradient of [Formula: see text] across [Formula: see text] grows linearly with [Formula: see text].

The relative Bruce–Roberts number of a function on a hypersurface

We consider the relative Bruce–Roberts number $\mu _{\textrm {BR}}^{-}(f,\,X)$ of a function on an isolated hypersurface singularity $(X,\,0)$ . We show that $\mu _{\textrm {BR}}^{-}(f,\,X)$ is equal to the sum of the Milnor number of the fibre $\mu (f^{-1}(0)\cap X,\,0)$ plus the difference $\mu (X,\,0)-\tau (X,\,0)$ between the Milnor and the Tjurina numbers of $(X,\,0)$ . As an application, we show that the usual Bruce–Roberts number $\mu _{\textrm {BR}}(f,\,X)$ is equal to $\mu (f)+\mu _{\textrm {BR}}^{-}(f,\,X)$ . We also deduce that the relative logarithmic characteristic variety $LC(X)^{-}$ , obtained from the logarithmic characteristic variety $LC(X)$ by eliminating the component corresponding to the complement of $X$ in the ambient space, is Cohen–Macaulay.

Characterization of the R2R3-MYB Transcription Factor CsMYB113 Regulates Anthocyanin Biosynthesis in Tea Plants (Camellia Sinensis)

Tea plant(Camellia sinensis) has very long history of cultivation and abundant germplasm resources in China. Purple bud is a characteristic variety, which has attracted the attention of breeding researchers because it accumulated a large number of anthocyanins naturally. In many species, R2R3-MYBtranscription factors (TFs)wereprovedto be involved in the regulation of anthocyanin biosynthesis.Research on anthocyanin metabolism has been relatively clear in some species, but that needs to be further elucidated in tea plants. In this research, anR2R3-MYB transcriptionfactor CsMYB113 relate to the anthocyanin accumulation regulation was identified from tea plants. Spatial and temporal expressionanalysis revealed differential expression of CsMYB113among different tissues and organs, with highest expression occurringin the roots.Subcellular localization assays showed that CsMYB113 localizedin the nucleus.Ectopic expression of CsMYB113increased pigmentation and anthocyanin contentsby the up-regulationof theexpression levelsof genes in anthocyanin biosynthesis pathwayamongdifferent tissues of Arabidopsis.Moreover, transient overexpressionof 35S::CsMYB113in tea plant increased the anthocyanin contents in the leaves.Our results indicated that CsMYB113 play important role in the anthocyaninbiosynthesis regulation in tea plants. It will also provide useful candidate gene for the modification of anthocyanin metabolism by genetic engineeringin plants.

Maps from Feigin and Odesskii's elliptic algebras to twisted homogeneous coordinate rings

The elliptic algebras in the title are connected graded $\mathbb {C}$ -algebras, denoted $Q_{n,k}(E,\tau )$ , depending on a pair of relatively prime integers $n>k\ge 1$ , an elliptic curve E and a point $\tau \in E$ . This paper examines a canonical homomorphism from $Q_{n,k}(E,\tau )$ to the twisted homogeneous coordinate ring $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ on the characteristic variety $X_{n/k}$ for $Q_{n,k}(E,\tau )$ . When $X_{n/k}$ is isomorphic to $E^g$ or the symmetric power $S^gE$ , we show that the homomorphism $Q_{n,k}(E,\tau ) \to B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ is surjective, the relations for $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ are generated in degrees $\le 3$ and the noncommutative scheme $\mathrm {Proj}_{nc}(Q_{n,k}(E,\tau ))$ has a closed subvariety that is isomorphic to $E^g$ or $S^gE$ , respectively. When $X_{n/k}=E^g$ and $\tau =0$ , the results about $B(X_{n/k},\sigma ',\mathcal {L}^{\prime }_{n/k})$ show that the morphism $\Phi _{|\mathcal {L}_{n/k}|}:E^g \to \mathbb {P}^{n-1}$ embeds $E^g$ as a projectively normal subvariety that is a scheme-theoretic intersection of quadric and cubic hypersurfaces.

Integrability via Geometry: Dispersionless Differential Equations in Three and Four Dimensions

We prove that the existence of a dispersionless Lax pair with spectral parameter for a nondegenerate hyperbolic second order partial differential equation (PDE) is equivalent to the canonical conformal structure defined by the symbol being Einstein–Weyl on any solution in 3D, and self-dual on any solution in 4D. The first main ingredient in the proof is a characteristic property for dispersionless Lax pairs. The second is the projective behaviour of the Lax pair with respect to the spectral parameter. Both are established for nondegenerate determined systems of PDEs of any order. Thus our main result applies more generally to any such PDE system whose characteristic variety is a quadric hypersurface.

Algebraic analysis of the hypergeometric function $$\,{_1F_{\!\!\;1}}\,$$ of a matrix argument

In this article, we investigate Muirhead’s classical system of differential operators for the hypergeometric function $$\,{_1F_{\!\!\;1}}\,$$ 1 F 1 of a matrix argument. We formulate a conjecture for the combinatorial structure of the characteristic variety of its Weyl closure which is both supported by computational evidence as well as theoretical considerations. In particular, we determine the singular locus of this system.

MEDICAL TREATMENT LANDSCAPE FOR ACTIVE ACROMEGALY IN A PITUITARY CENTER IN ISRAEL

Objective: To evaluate current real-life experience with medical treatment for active acromegaly in a large cohort. Methods: Data on demographic parameters, blood tests, imaging studies, and treatments were extracted from the medical records. Results: The cohort included 87 patients (43 male) with active acromegaly. The mean age at diagnosis was 40.2 ± 11.4 years, and the mean duration of follow-up was 7.9 ± 5.8 years. Seventy patients presented with a macroadenoma. Mean baseline insulin growth factor 1 (IGF-1) (n = 67) was 3.2 ± 1.9 × upper limit of normal (ULN). Surgery and radiotherapy were performed in 75 and 10 patients, respectively. Currently, 38 subjects receive somatostatin analogues, pegvisomant as a monotherapy is given to 8 patients, pasireotide is given to 17 patients, cabegoline to 4 patients, estrogen to 2 females, and SSAs combined with pegvisomant to 10 patients. Eight patients are not being actively treated, including 4 following radiotherapy. Good biochemical control (IGF-1 <1.3 × ULN) was achieved in 76 patients (87%), and 11 patients (13%) are currently uncontrolled (IGF-1 >1.3 × ULN). Seventy-eight percent of controlled patients are being given 1 medication; 11% are on combination therapy; 4 patients are well controlled after radiotherapy and 2 are partially controlled without any treatment. The main adverse effects of treatment were diabetes mellitus in 7 patients (on pasireotide) and symptomatic cholelithiasis in 5 patients. Conclusion: Active acromegaly can be controlled medically in most patients, with a low rate of adverse effects. This study displays the characteristic variety of treatment options available for active acromegaly. Abbreviations: DM = diabetes mellitus; IGF-1 = insulin-like growth factor 1; LAR = long-acting release; SSA = somatostatin analogues; ULN = upper limit of normal

Mesonephric-Like Adenocarcinoma: Two Cases Of A Rare Entity And Review Of Literature

Introduction/Objective Mesonephric-like adenocarcinoma (MLA) is a rare gynecologic malignancy occurring in the uterus and ovary and is morphologic similar to cervical mesonephric adenocarcinoma. We present two cases of mesonephric-like adenocarcinoma of the ovary. Methods The first case is a 72-year-old female presented with abdominal pain with bloating. Abdominal CT scan showed a 7.4 x 6.3 x 6.1 cm heterogenous right adnexal mass. The second case is a 66-year-old female with history of heavy bleeding and endometriosis, status post total abdominal hysterectomy with bilateral salpingo-oophorectomy 20 years ago. Abdominal CT scan demonstrated a 5.3 cm right pelvic mass and 12.5 cm right pelvic sidewall lymph node. Results Both tumors have characteristic variety of features including ductal, tubular, retiform, papillary, solid and infiltrative morphological patterns. Cytologic atypia is mild to moderate. The histological features can raise a broad differential diagnosis including primary tubo-ovarian carcinomas, sex cord-stromal tumors, and metastatic carcinomas. Immunohistochemical stains demonstrate the ovarian tumor cells are positive for multiple pan-epithelial markers (AE1/AE3); negative for sex cord-stromal markers (inhibin and calretinin); positive for PAX8, GATA3, TTF1, and CD10; and negative for ER, PR, and WT1. In the first case, a small incidental endometrial carcinoma arising from an endometrial polyp was also identified. Morphologically, it is distinct from the ovarian mesonephric-like adenocarcinoma. Conclusion Morphological and immunohistochemical features of the two tumors are consistent with MLA, a very rare ovarian tumor that is thought to arise either from mesonephric remnants in the para-ovarian tissue or by trans- differentiation from a benign or neoplastic Mullerian precursor. These tumors are reported to have a very high frequency of KRAS mutations followed by PIK3CA mutations. Due to their rarity, the clinical behavior of these tumors is unknown.

Some computations on the characteristic variety of a line arrangement

We find monodromy formulas for line arrangements that are fibered with respect to the projection from one point. We use them to find 0-dimensional translated components in the first characteristic variety of the arrangement "Equation missing" determined by a regular n-polygon and its diagonals. We also find new 1-dimensional translated components which generalize the well-known case of the $$B_3$$ B 3 -deleted arrangement.

The Bruce–Roberts Number of A Function on A Hypersurface with Isolated Singularity

Let $(X,0)$ be an isolated hypersurface singularity defined by $\phi \colon ({\mathbb{C}}^n,0)\to ({\mathbb{C}},0)$ and $f\colon ({\mathbb{C}}^n,0)\to{\mathbb{C}}$ such that the Bruce–Roberts number $\mu _{BR}(f,X)$ is finite. We first prove that $\mu _{BR}(f,X)=\mu (f)+\mu (\phi ,f)+\mu (X,0)-\tau (X,0)$, where $\mu $ and $\tau $ are the Milnor and Tjurina numbers respectively of a function or an isolated complete intersection singularity. Second, we show that the logarithmic characteristic variety $LC(X,0)$ is Cohen–Macaulay. Both theorems generalize the results of a previous paper by some of the authors, in which the hypersurface $(X,0)$ was assumed to be weighted homogeneous.