SOME REMARKS ON THE SMARANDACHE FUNCTION

Remarks on a Function in the Number Theory

Secondary giant cell glioblastoma in a multiple drug abuser - simple association or ethiopathogenic correlation? Case presentation and literature review

Experimental investigations have shown that drug abuse initiates a cascade of pathophysiological events including toxic and hypoxic-ischemic injury on neurons, microglia and astrocytes, which finally lead to widespread disturbances in the brain. There are many reports about the psychiatric and neurologic effects of multiple drug abuse, but only a few clinical studies reporting possible correlation between recreational illicit drugs and gliomas. In this paper we present the case of a 40 years-old male patient, with a long history (almost ten years) of multiple drug abuse, including cocaine, heroin, marijuana, ethnobotanical drugs and nicotine, who was diagnosed and surgically treated for a supratentorial secondary giant cell glioblastoma (sgcGB) developed in a diffuse astrocytoma NOS. Depending on the type of the illicit drug used by the patient and the moment of life he used them, the morphological features identified in the histological samples of our patient confirmed the gliomagenesis effect of chronic multiple drug abuse, but also its inhibitory effects on tumour cells growth. This was due to the fact that although the tumour was large in size and caused brain sub-falcine herniation, the patient reported the onset of seizures only late in the evolution. In conclusion, the diagnosis of a brain tumour should take into consideration not only patient's clinical and imaging data, but also his lifestyle, especially his addiction to recreational drugs

A monotonicity property of Riemann's xi function and a reformulation of the Riemann Hypothesis

We prove that Riemann's xi function is strictly increasing (respectively, strictly decreasing) in modulus along every horizontal half-line in any zero-free, open right (respectively, left) half-plane. A corollary is a reformulation of the Riemann Hypothesis.Comment: 4 pages, published versio

Thermodynamic signatures for the existence of Dirac electrons in ZrTe5

We combine transport, magnetization, and torque magnetometry measurements to investigate the electronic structure of ZrTe5 and its evolution with temperature. At fields beyond the quantum limit, we observe a magnetization reversal from paramagnetic to diamagnetic response, which is characteristic of a Dirac semi-metal. We also observe a strong non-linearity in the magnetization that suggests the presence of additional low-lying carriers from other low-energy bands. Finally, we observe a striking sensitivity of the magnetic reversal to temperature that is not readily explained by simple band-structure models, but may be connected to a temperature dependent Lifshitz transition proposed to exist in this material.Comment: 5 pages, 4 figure

From the Hitchin section to opers through nonabelian Hodge

For a complex simple simply connected Lie group $G$, and a compact Riemann surface $C$, we consider two sorts of families of flat $G$-connections over $C$. Each family is determined by a point ${\mathbf u}$ of the base of Hitchin's integrable system for $(G,C)$. One family $\nabla_{\hbar,{\mathbf u}}$ consists of $G$-opers, and depends on $\hbar \in {\mathbb C}^\times$. The other family $\nabla_{R,\zeta,{\mathbf u}}$ is built from solutions of Hitchin's equations, and depends on $\zeta \in {\mathbb C}^\times, R \in {\mathbb R}^+$. We show that in the scaling limit $R \to 0$, $\zeta = \hbar R$, we have $\nabla_{R,\zeta,{\mathbf u}} \to \nabla_{\hbar,{\mathbf u}}$. This establishes and generalizes a conjecture formulated by Gaiotto