Mean Value Theorems for Binary Egyptian Fractions

In this paper, we establish two mean value theorems for the number of solutions of the Diophantine equation $\frac{a}{n}=\frac{1}{x}+\frac{1}{y}$, in the case when $a$ is fixed and $n$ varies and in the case when both $a$ and $n$ vary.Comment: 16 pages, to appear in J. Number Theor

On the characteristics of emulsion chamber family events produced in low heights

The uncertainty of the primary cosmic ray composition at 10 to the 14th power -10 to the 16th power eV is well known to make the study of the nuclear interaction mechanism more difficult. Experimentally considering, if one can identify effectively the family events which are produced in low heights, then an event sample induced by primary protons might be able to be separated. It is undoubtedly very meaningful. In this paper an attempt is made to simulate the family events under the condition of mountain emulsion chamber experiments with a reasonable model. The aim is to search for the dependence of some experimentally observable quantities to the interaction height

Network Properties Revealed during Multi-Scale Calcium Imaging of Seizure Activity in Zebrafish.

Seizures are characterized by hypersynchronization of neuronal networks. Understanding these networks could provide a critical window for therapeutic control of recurrent seizure activity, i.e., epilepsy. However, imaging seizure networks has largely been limited to microcircuits in vitro or small "windows" in vivo. Here, we combine fast confocal imaging of genetically encoded calcium indicator (GCaMP)-expressing larval zebrafish with local field potential (LFP) recordings to study epileptiform events at whole-brain and single-neuron levels in vivo. Using an acute seizure model (pentylenetetrazole, PTZ), we reliably observed recurrent electrographic ictal-like events associated with generalized activation of all major brain regions and uncovered a well-preserved anterior-to-posterior seizure propagation pattern. We also examined brain-wide network synchronization and spatiotemporal patterns of neuronal activity in the optic tectum microcircuit. Brain-wide and single-neuronal level analysis of PTZ-exposed and 4-aminopyridine (4-AP)-exposed zebrafish revealed distinct network dynamics associated with seizure and non-seizure hyperexcitable states, respectively. Neuronal ensembles, comprised of coactive neurons, were also uncovered during interictal-like periods. Taken together, these results demonstrate that macro- and micro-network calcium motifs in zebrafish may provide a greater understanding of epilepsy

Gravity in a Box

We consider a brane-world construction which incorporates a finite region of flat space, ``the box,'' surrounded by a region of anti-de Sitter space. This hybrid construction provides a framework which interpolates between the scenario proposed by Arkani-Hamed, Dimopoulos and Dvali, and that proposed by Randall and Sundrum. Within this composite framework, we investigate the effects of resonant modes on four-dimensional gravity. We also show that, on a probe brane in the anti-de Sitter region, there is enhanced production of on-shell nonresonant modes. We compare our model to some recent attempts to incorporate the Randall-Sundrum scenario into superstring theory.Comment: 15 pages, 1 figur