Ham Sandwich is Equivalent to Borsuk-Ulam

The Borsuk-Ulam theorem is a fundamental result in algebraic topology, with applications to various areas of Mathematics. A classical application of the Borsuk-Ulam theorem is the Ham Sandwich theorem: The volumes of any n compact sets in R^n can always be simultaneously bisected by an (n-1)-dimensional hyperplane. In this paper, we demonstrate the equivalence between the Borsuk-Ulam theorem and the Ham Sandwich theorem. The main technical result we show towards establishing the equivalence is the following: For every odd polynomial restricted to the hypersphere f:S^n->R, there exists a compact set A in R^{n+1}, such that for every x in S^n we have f(x)=vol(A cap H^+) - vol(A cap H^-), where H is the oriented hyperplane containing the origin with x as the normal. A noteworthy aspect of the proof of the above result is the use of hyperspherical harmonics. Finally, using the above result we prove that there exist constants n_0, epsilon_0>0 such that for every n>= n_0 and epsilon <= epsilon_0/sqrt{48n}, any query algorithm to find an epsilon-bisecting (n-1)-dimensional hyperplane of n compact set in [-n^4.51,n^4.51]^n, even with success probability 2^-Omega(n), requires 2^Omega(n) queries

Quiescence of an Outburst of a Low-Mass Young Stellar Object: LDN1415-IRS

LDN1415-IRS, a low-mass young stellar object (YSO) went into an outburst between 2001 and 2006, illuminating a surrounding nebula, LDN1415-Neb. LDN1415-Neb was found to have brightened by I=3.77 mag by April 2006. The optical light curve covering $\sim$ 15.5 years, starting from October 2006 to January 2022, is presented in this study. The initial optical spectrum indicated the presence of winds in the system but the subsequent spectra have no wind indicators. The declining light curve and the absence of the P-Cygni profile in later epoch spectra indicate that the star and nebula system is retrieving back from its outburst state. Two Herbig-Haro objects (HHOs) are positioned linearly with respect to the optical brightness peak of the nebula, probably indicating the circumstellar disk being viewed edge-on. Our recent deep near-infrared (NIR) imaging using TANSPEC has revealed the presence of a nearby star-like source, south of the LDN1415-IRS, at an angular distance of $\sim$ 5.4 arcsec.Comment: Accepted for publication in the Journal of Astrophysics and Astronom

Post Traumatic Glioma: An association questioned

Post traumatic glioma has been a matter of debate. Few reports favor its occurrence in previous head injury scar, subsequently developing into glioma. Here we report a case of young patient presented with headache, seizures and gradual loss of vision. On investigation found to have brain tumor. Patient had head injury occurred 3 yrs back. It fulfills all the criteria required to establish traumatic origin, further supporting the association

Post-outburst evolution of bonafide FUor V2493 Cyg: A Spectro-photometric monitoring

We present here the results of eight years of our near-simultaneous optical/near-infrared spectro-photometric monitoring of bonafide FUor candidate `V2493 Cyg' starting from 2013 September to 2021 June. During our optical monitoring period (between October 16, 2015 and December 30, 2019), the V2493 Cyg is slowly dimming with an average dimming rate of $\sim$26.6 $\pm$ 5.6 mmag/yr in V band. Our optical photometric colors show a significant reddening of the source post the second outburst pointing towards a gradual expansion of the emitting region post the second outburst. The mid infra-red colors, on the contrary, exhibits a blueing trend which can be attributed to the brightening of the disc due to the outburst. Our spectroscopic monitoring shows a dramatic variation of the H$\alpha$ line as it transitioned from absorption feature to the emission feature and back. Such transition can possibly be explained by the variation in the wind structure in combination with accretion. Combining our time evolution spectra of the Ca II infra-red triplet lines with the previously published spectra of V2493 Cyg, we find that the accretion region has stabilised compared to the early days of the outburst. The evolution of the O I $\lambda$7773 \AA~ line also points towards the stabilization of the circumstellar disc post the second outburst.Comment: 34 pages, 12 figures, 6 tables, accepted for publication in Ap