A Survey on the Ternary Purely Exponential Diophantine Equation ax+by=cza^x + b^y = c^z

Let $a$, $b$, $c$ be fixed coprime positive integers with $\min\{a,b,c\}>1$. In this survey, we consider some unsolved problems and related works concerning the positive integer solutions $(x,y,z)$ of the ternary purely exponential diophantine equation $a^x + b^y = c^z$

Words are Malleable: Computing Semantic Shifts in Political and Media Discourse

Recently, researchers started to pay attention to the detection of temporal shifts in the meaning of words. However, most (if not all) of these approaches restricted their efforts to uncovering change over time, thus neglecting other valuable dimensions such as social or political variability. We propose an approach for detecting semantic shifts between different viewpoints--broadly defined as a set of texts that share a specific metadata feature, which can be a time-period, but also a social entity such as a political party. For each viewpoint, we learn a semantic space in which each word is represented as a low dimensional neural embedded vector. The challenge is to compare the meaning of a word in one space to its meaning in another space and measure the size of the semantic shifts. We compare the effectiveness of a measure based on optimal transformations between the two spaces with a measure based on the similarity of the neighbors of the word in the respective spaces. Our experiments demonstrate that the combination of these two performs best. We show that the semantic shifts not only occur over time, but also along different viewpoints in a short period of time. For evaluation, we demonstrate how this approach captures meaningful semantic shifts and can help improve other tasks such as the contrastive viewpoint summarization and ideology detection (measured as classification accuracy) in political texts. We also show that the two laws of semantic change which were empirically shown to hold for temporal shifts also hold for shifts across viewpoints. These laws state that frequent words are less likely to shift meaning while words with many senses are more likely to do so.Comment: In Proceedings of the 26th ACM International on Conference on Information and Knowledge Management (CIKM2017

Merger as Intermittent Accretion

The Self-Similar Secondary Infall Model (SSIM) is modified to simulate a merger event. The model encompass spherical versions of tidal stripping and dynamical friction that agrees with the Syer & White merger paradigm's behaviour. The SSIM shows robustness in absorbing even comparable mass perturbations and returning to its original state. It suggests the approach to be invertible and allows to consider accretion as smooth mass inflow merging and mergers as intermittent mass inflow accretion.Comment: letter accepted by A&A 29/09/08, 4 pages, colour figure

Microstructural parameter estimation in vivo using diffusion MRI and structured prior information.

Diffusion MRI has recently been used with detailed models to probe tissue microstructure. Much of this work has been performed ex vivo with powerful scanner hardware, to gain sensitivity to parameters such as axon radius. By contrast, performing microstructure imaging on clinical scanners is extremely challenging

Lingual Salt Glands in Crocodylus acutus and C. johnstoni and Their Absence from Alligator mississipiensis and Caiman crocodilus

1. Lingual salt glands, secreting hyperosmotic Na/K solutions in response to methacholine, are present in Crocodylus acutus and C. johnstoni but apparently absent from the alligatorids, Alligator mississipiensis and Caiman crocodilus. 2. Both secretory rates (6-20 [micro-mol/100 g-h) and concentrations (450-600 mM Na) of glandular secretions are essentially identical in the marine/estuarine C. acutus and C. porosus and significantly higher than in the freshwater C. johnstoni (1-2 micro-mol/100 g-h; 320-420 mM Na). 3. Lingual glands in Alligator secrete isosmotic Na/K at low rates (1-2 micro-mol/100 g-h) while those of Caiman show no response to methacholine. 4. The physiological contrast between alligatorids and crocodylids is reflected in distinct differences in the superficial appearance of the tongue and lingual pores. 5. It is postulated that the alligatorid condition of low secretory capacity and isosmotic secretion reflects the primitive salivary function of lingual glands from which the salt-secreting capability in crocodylids was derived

On the universality of density profiles

We use the secondary infall model described in Del Popolo (2009), which takes into account the effect of dynamical friction, ordered and random angular momentum, baryons adiabatic contraction and dark matter baryons interplay, to study how in- ner slopes of relaxed LCDM dark matter (DM) halos with and without baryons (baryons+DM, and pure DM) depend on redshift and on halo mass. We apply the quoted method to structures on galactic scales and clusters of galaxies scales. We find that the inner logarithmic density slope, of dark matter halos with baryons has a significant dependence on halo mass and redshift with slopes ranging from 0 for dwarf galaxies to 0.4 for objects of M = 10^13M_solar and 0.94 for M = 10^15M_solar clusters of galaxies. Structures slopes increase with increasing redshift and this trend reduces going from galaxies to clusters. In the case of density profiles constituted just of dark matter the mass and redshift dependence of slope is very slight. In this last case, we used the Merrit et al. (2006) analysis who compared N-body density profiles with various parametric models finding systematic variation in profile shape with halo mass. This last analysis suggests that the galaxy-sized halos obtained with our model have a different shape parameter, i.e. a different mass distribution, than the cluster-sized halos, obtained with the same model. The results of the present paper argue against universality of density profiles constituted by dark matter and baryons and confirm claims of a systematic variation in profile shape with halo mass, for dark matter halos.Comment: 11 pages, 5 figure

The effects of primordial non-Gaussianity on giant-arc statistics

For over a decade, it has been debated whether the concordance LCDM model is consistent with the observed abundance of giant arcs in clusters. While previous theoretical studies have focused on properties of the lens and source populations, as well as cosmological effects such as dark energy, the impact of initial conditions on the giant-arc abundance is relatively unexplored. Here, we quantify the impact of non-Gaussian initial conditions with the local bispectrum shape on the predicted frequency of giant arcs. Using a path-integral formulation of the excursion set formalism, we extend a semi-analytic model for calculating halo concentrations to the case of primordial non-Gaussianity, which may be useful for applications outside of this work. We find that massive halos tend to collapse earlier in models with positive f_NL, relative to the Gaussian case, leading to enhanced concentration parameters. The converse is true for f_NL < 0. In addition to these effects, which change the lensing cross sections, non-Gaussianity also modifies the abundance of supercritical clusters available for lensing. These combined effects work together to either enhance (f_NL > 0) or suppress (f_NL < 0) the probability of giant-arc formation. Using the best value and 95% confidence levels currently available from the Wilkinson Microwave Anisotropy Probe, we find that the giant-arc optical depth for sources at z_s~2 is enhanced by ~20% and ~45% for f_NL = 32 and 74 respectively. In contrast, we calculate a suppression of ~5% for f_NL = -10. These differences translate to similar relative changes in the predicted all-sky number of giant arcs.Comment: 16 pages, 8 figures, accepted by MNRA

Cosmic shear statistics in cosmologies with non-Gaussian initial conditions

We computed the power spectrum of weak cosmic shear in models with non-Gaussian primordial density fluctuations. Cosmological initial conditions deviating from Gaussianity have recently attracted much attention in the literature, especially with respect to their effect on the formation of non-linear structures and because of the bounds that they can put on the inflationary epoch. The fully non-linear matter power spectrum was evaluated with the use of the physically motivated, semi-analytic halo model, where the mass function and linear halo bias were suitably corrected for non-Gaussian cosmologies. In agreement with previous work, we found that a level of non-Gaussianity compatible with CMB bounds and with positive skewness produces an increase in power of the order of a few percent at intermediate scales. We then used the matter power spectrum, together with observationally motivated background source redshift distributions in order to compute the cosmological weak lensing power spectrum. We found that the degree of deviation from the power spectrum of the reference Gaussian model is small compared to the statistical error expected from even future weak lensing surveys. However, summing the signal over a large range of multipoles can beat down the noise, bringing to a significant detection of non-Gaussianity at the level of $|f_\mathrm{NL}| \simeq$ few tens, when all other cosmological parameters are held fixed. Finally, we have shown that the constraints on the level of non-Gaussianity can be improved by $\sim 20$ with the use of weak lensing tomography.Comment: 15 pages, 10 figures. Accepted by MNRA

On a conjecture concerning the number of solutions to ax+by=cza^x+b^y=c^z, II

Let $a$, $b$, $c$ be fixed coprime positive integers with $\min\{ a,b,c \} >1$, $a<b$. Let $N(a,b,c)$ denote the number of positive integer solutions $(x,y,z)$ of the equation $a^x + b^y = c^z$. We show that if $(a,b,c)$ is a triple of distinct primes for which $N(a,b,c)>1$ and $(a,b,c)$ is not one of the six known such triples then we must have $a=2$, $b \equiv 1 \bmod 48$, $c \equiv 17 \mod 48$, $2 \mid v_2(p-1) \le v_2(q-1)$ where $v_2(n)$ satisfies $2^{v_2(n)} \parallel n$, at least one of the multiplicative orders $u_q(p)$ or $u_p(q)$ must be odd where $u_p(n)$ is the least integer $t$ such that $n^t \equiv 1 \bmod p$, and 2 must be an octic residue modulo $q$ except for two specific cases. These results support a conjecture put forward in [26] and improve results in [15]

The baryonic content and Tully-Fisher relation at z~0.6

[abr.] Using the multi-integral-field spectrograph GIRAFFE at VLT, we previsouly derived the stellar-mass Tully-Fisher Relation (smTFR) at z~0.6, and found that the distant relation is systematically offset by roughly a factor of two toward lower masses. We extend the study of the evolution of the TFR by establishing the first distant baryonic TFR. To derive gas masses in distant galaxies, we estimate a gas radius and invert the Schmidt-Kennicutt law between star formation rate and gas surface densities. We find that gas extends farther out than the UV light from young stars, a median of ~30%. We present the first baryonic TFR (bTFR) ever established at intermediate redshift and show that, within an uncertainty of +/-0.08 dex, the zeropoint of the bTFR does not appear to evolve between z~0.6 and z=0. The absence of evolution in the bTFR over the past 6 Gyr implies that no external gas accretion is required for distant rotating disks to sustain star formation until z=0 and convert most of their gas into stars. Finally, we confirm that the larger scatter found in the distant smTFR, and hence in the bTFR, is caused entirely by major mergers. This scatter results from a transfer of energy from bulk motions in the progenitors, to random motions in the remnants, generated by shocks during the merging. Shocks occurring during these events naturally explain the large extent of ionized gas found out to the UV radius in z~0.6 galaxies. All the results presented in this paper support the ``spiral rebuilding scenario'' of Hammer and collaborators, i.e., that a large fraction of local spiral disks have been reprocessed during major mergers in the past 8 Gyr.Comment: Accepted for publication in A&A, v3 addressing comments from the refere