Abstract:
Define a body A to be able to hide behind a body B if the orthogonal projection of B contains a translation of the corresponding orthogonal projection of A in every direction. In two dimensions, it is easy to observe that there exist two objects such that one can hide behind another and have a larger area than the other. It was recently shown that similar examples exist in higher dimensions as well. However, the highest possible volume ratio for such bodies is still undetermined. We investigated two three-dimensional examples, one involving a tetrahedron and a ball and the other involving a tetrahedron and an inverted tetrahedron. We calculate the highest volume ratio known up to this date, 1.16, which is generated by our second example.

Abstract:
We present English translation of the classical article of Hermann Amadeus Schwarz (1843--1921) "Proof of the theorem that a surface area of a ball is smaller than of any other body of the same volume" which was published in 1884, in Proceedings of the K"onigliche Gesellschaft der Wissenschaften and the Georg-Augusts-Universit"at, G"ottingen. We preserved the author notations throughout the text and tried to follow his grammar construction of the sentences.

Abstract:
Telomere length shortens with cellular division, and leukocyte telomere length is used as a marker for systemic telomere length. The hippocampus hosts adult neurogenesis and is an important structure for episodic memory, and carriers of the apolipoprotein E ε4 allele exhibit higher hippocampal atrophy rates and differing telomere dynamics compared with non-carriers. The authors investigated whether leukocyte telomere length was associated with hippocampal volume in 57 cognitively intact subjects (29 ε3/ε3 carriers; 28 ε4 carriers) aged 49–79 yr. Leukocyte telomere length correlated inversely with left (rs = ？0.465; p = 0.011), right (rs = ？0.414; p = 0.025), and total hippocampus volume (rs = ？0.519; p = 0.004) among APOE ε3/ε3 carriers, but not among ε4 carriers. However, the ε4 carriers fit with the general correlation pattern exhibited by the ε3/ε3 carriers, as ε4 carriers on average had longer telomeres and smaller hippocampi compared with ε3/ε3 carriers. The relationship observed can be interpreted as long telomeres representing a history of relatively low cellular proliferation, reflected in smaller hippocampal volumes. The results support the potential of leukocyte telomere length being used as a biomarker for tapping functional and structural processes of the aging brain.

Abstract:
The best discussion of that situation I have ever read is over 2000 years old. It's Plato's Allegory of the Cave, and it's one of my favorite passages in classical literature.The allegory is presented as an imaginary dialogue between Socrates and Plato's brother Glaucon, but it's really Plato speaking. Imagine, he says, a group of people who are born and live all their lives in a cave. They are forced to sit in chairs facing the back wall of the cave, restrained so that they cannot look anywhere else. Behind them, at the mouth of the cave, is a large fire, and between them and the fire is a walkway along which people carrying things, including replicas of animals, pass continuously. All the people of the cave can ever see are the shadows cast on the wall in front of them by those passing behind. All they can hear are the echoes in the cave produced by the movements they never see. Would they not, Plato asks, come to believe that those shadows and echoes are reality? Would they not assume that the entire world consists of the cave and the shadows on the wall? Wouldn't they praise as clever whoever could best guess which shadow would come next as someone who understood the nature of the world? And wouldn't the whole of their society come to depend on the shadows on the wall?It's a powerful image, but Plato takes it further. Now let us suppose, he says, that one of the people of the cave is freed from his chair and allowed to face the outside. Would he not first be blinded by the fire? And then, as his eyes adapted and he saw the people passing by on the walkway, would he not distrust the things he saw, believing that his eyes deceived him, because what they showed him contradicted what he knew reality had to be?Then Plato goes still one step more. Let us now imagine that our freed cave dweller eventually acclimates to the world outside the cave, and recognizes that as reality. 'Wouldn't he then remember his first home, what passed for wisdom there, and his fellow pris

Abstract:
Using backwards ray tracing, we study the shadows of Kerr black holes with scalar hair (KBHsSH). KBHsSH interpolate continuously between Kerr BHs and boson stars (BSs), so we start by investigating the lensing of light due to BSs. Moving from the weak to the strong gravity region, BSs - which by themselves have no shadows - are classified, according to the lensing produced, as: $(i)$ non-compact, which yield no multiple images; $(ii)$ compact, which produce an increasing number of Einstein rings and multiple images of the whole celestial sphere; $(iii)$ ultra-compact, which possess light rings, yielding an infinite number of images with (we conjecture) a self-similar structure. The shadows of KBHsSH, for Kerr-like horizons and non-compact BS-like hair, are analogous to, but distinguishable from, those of comparable Kerr BHs. But for non-Kerr-like horizons and ultra-compact BS-like hair, the shadows of KBHsSH are drastically different: novel shapes arise, sizes are considerably smaller and multiple shadows of a single BH become possible. Thus, KBHsSH provide quantitatively and qualitatively new templates for ongoing (and future) very large baseline interferometry (VLBI) observations of BH shadows, such as those of the Event Horizon Telescope.

Abstract:
The Ekeland variational principle implies what can be regarded as a strong version, in the $C^1$ category, of the Yau minimum principle: under the appropriate hypotheses {\it every} minimizing sequence admits a {\it good shadow}, a second minimizing sequence that has good properties and is asymptotic to the original one. Using arguments from dynamical systems, we give another proof of this result and also establish, with the aid of Gromov's theorem on monotonicity of volume ratios, a special case of a conjecture claiming the existence of good shadows in the original $C^2$ setting of the Yau minimum principle. The interest in having an abundance of good shadows stems from the fact that this is a desirable property if one wants to refine the applications of the asymptotic minimum principle, as it allows for information to be localized at infinity. These ideas are applied in this paper to the study of the convex hulls of complete submanifolds of Euclidean $n$-space that have controlled Grassmanian-valued Gauss maps.

Abstract:
We define new natural variants of the notions of weighted covering and separation numbers and discuss them in detail. We prove a strong duality relation between weighted covering and separation numbers and prove a few relations between the classical and weighted covering numbers, some of which hold true without convexity assumptions and for general metric spaces. As a consequence, together with some volume bounds that we discuss, we provide a bound for the famous Levi-Hadwiger problem concerning covering a convex body by homothetic slightly smaller copies of itself, in the case of centrally symmetric convex bodies, which is qualitatively the same as the best currently known bound. We also introduce the weighted notion of the Levi-Hadwiger covering problem, and settle the centrally-symmetric case, thus also confirm Nasz\'{o}di's equivalent fractional illumination conjecture in the case of centrally symmetric convex bodies (including the characterization of the equality case, which was unknown so far).

Abstract:
The Xapagy architecture is a story-oriented cognitive system which relies exclusively on the autobiographical memory implemented as a raw collection of events. Reasoning is performed by shadowing current events with events from the autobiography. The shadows are then extrapolated into headless shadows (HLSs). In a story following mood, HLSs can be used to track the level of surprise of the agent, to infer hidden actions or relations between the participants, and to summarize ongoing events. In recall mood, the HLSs can be used to create new stories ranging from exact recall to free-form confabulation.

Abstract:
The eigenvalue structure of the quantum transfer matrix is known to encode essential information about the elementary excitations. Here we study transfer matrices of quantum states in a topological phase using the tensor network formalism. We demonstrate that topological quantum order requires a particular type of `symmetry breaking' for the fixed point subspace of the transfer matrix, and relate physical anyon excitations to domain wall excitations at the level of the transfer matrix. A topological phase transition to a trivial phase triggers a change in the fixed point subspace to either a larger or smaller symmetry and we explain how this relates to a condensation or confinement of the corresponding anyon sectors. The tensor network formalism enables us to determine the structure of the topological sectors in two-dimensional gapped phases very efficiently, therefore opening novel avenues for studying fundamental open questions related to anyon condensation.

Abstract:
Spectrahedral shadows are projections of linear sections of the cone of positive semidefinite matrices. We characterize the polynomials that vanish on the boundaries of these convex sets when both the section and the projection are generic.