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Let $(M,g)$ be a compact smooth connected Riemannian manifold (without boundary) of dimension $N\ge7$. Assume $M$ is symmetric with respect to a point $ξ_0$ with non-vanishing Weyl's tensor. We consider the linear perturbation of the Yamabe problem $$(P_ε)\qquad-\mathcal L_g u+εu=u^{N+2\over N-2}\ \hbox{in}\ (M,g) .$$ We prove that for any $k\in \mathbb N$, there exists $ε_k>0$ such that for all $ε\in (0, ε_k)$ the problem $(P_ε)$ has a symmetric solution $u_ε,$ which looks like the superposition of $k$ positive bubbles centered at the point $ξ_0$ as $ε\to 0$. In particular, $ξ_0$ is a {\em towering} blow-up point.
Scientists studying magma from the 2021 Tajogaite eruption on La Palma have discovered that extreme heat can dramatically change how an eruption unfolds。 When magma becomes superheated, it can dissolve the tiny crystal seeds that normally trigger crystallization, allowing the molten rock to remain fluid far longer as it rises
We develop a bottom-up framework to reconstruct the asymptotic spectrum of light towers in four-dimensional $\mathcal{N}=1$ effective field theories directly from their Kähler potential. Our construction uses the Integral Scaling Relation, which relates the mass scales of towers becoming light at infinite distance to the tensions of EFT strings, together with the Emergent String Conjecture applied recursively upon decompactification. These conditions organize the tower scaling vectors into a lattice generated by the EFT-string vectors and select the globally consistent tower polytopes. When applied to Kähler potentials arising from string compactifications, our algorithm precisely reproduces the known arrangements of towers and duality frames. We then classify the admissible polytopes for asymptotic Kähler potentials $K\sim-\log P(s)$, with $P(s)$ a homogeneous polynomial of degree at most seven, and show that certain apparently consistent Kähler potentials are incompatible with the assumed quantum-gravity constraints. For general polynomials, additional restrictions arise from gluing the tower arrangements across different growth sectors. Remarkably, every tower polytope allowed b
We prove that every perfectoid tower can be realized as the fiber product of a diagram involving perfectoid towers that are either $p$-torsion free or perfect of characteristic $p$. As an application, we conclude that separated perfectoid towers are reduced. We also establish the tilting invariance of étale cohomology and Koszul homology for perfectoid towers. As further applications, we prove that tilting preserves fundamental properties of Noetherian local rings such as being Cohen--Macaulay, Gorenstein, complete intersection, or regular.
To connect arithmetic and ring-theoretic properties of rings of mixed characteristic with those of positive characteristic, we introduce monoidal maps for perfectoid towers. Using these maps, we discuss the almost integrality of perfectoid towers and of their tilts. We also show that the towers constructed by F. Andreatta via ramification theory become perfectoid towers, and we apply the monoidal maps to deduce the normality of their small tilts.
We prove a mixed-characteristic analogue of Kunz's theorem in terms of perfectoid towers: a Noetherian local ring of residue characteristic $p$ is regular if and only if it admits a flat map to a Noetherian ring that extends to a perfectoid tower. This result is deduced from another mixed-characteristic analogue due to O. Gabber and J. Lurie. We also characterize regularity for perfectoid towers via vanishing of single higher $\mathrm{Tor}$-module of the residue field with a perfectoid algebra.
The main result of this paper shows that a weak form of Tower Sealing holds in a generic extension of hod mice with a strong cardinal and a proper class of Woodin cardinals. We show Tower Sealing fails in such extensions in general. We show that this weak form of Tower Sealing (called Partial Tower Sealing) implies Sealing and that its consistency strength is below that of ZFC + there is a Woodin limit of Woodin cardinals.
We present a unified construction of perfectoid towers from specific prisms which covers all the previous constructions of (p-torsion-free) perfectoid towers. By virtue of the construction, perfectoid towers can be systematically constructed for a large class of rings with Frobenius lift. Especially, any Frobenius lifting of a reduced $\mathbb{F}_p$-algebra has a perfectoid tower.
In this paper, we proved that there exist four distinct diffeomorphism classes of three-dimensional real Bott tower $M(A)=(S^1)^3/(\mathbb{Z}_2)^3$, and 12 distinct diffeomorphism classes of four-dimensional real Bott tower $M(A)=(S^1)^4/(\mathbb{Z}_2)^4$, where matrix $A$ corresponds to the action of $(\mathbb{Z}_2)^n$ on $(S^1)^n$ for n=3,4.
The weighted Tower of Hanoi is a new generalization of the classical Tower of Hanoi problem, where a move of a disc between two pegs $i$ and $j$ is weighted by a positive real $w_{ij}\geq 0$. This new problem generalizes the concept of finding the minimum number of moves to solve the Tower of Hanoi, to find a sequence of moves with the minimum total cost. We present an optimal dynamic algorithm to solve the weighted Tower of Hanoi problem, we also establish some properties of this problem, as well as its relation with the Tower of Hanoi variants that are based on move restriction.
Sony doesn't think the move will hurt it financially
Sweden could become an important source of rare earth elements needed for magnets and green technologies。 Instead of extracting one target metal, researchers are cataloging everything contained in Swedish mineral deposits。 They will then develop magnets using those naturally available combinations, potentially reducing pollution, waste, and relianc
NASA astronaut Chris Williams and two Roscosmos cosmonauts are back on Earth after spending 241 days aboard the International Space Station。 Their journey covered more than 102 million miles and included 3,856 trips around the planet
Europa’s hidden ocean has made the icy moon one of the solar system’s most promising places to search for habitable conditions。 Scientists have hoped that water from this deep ocean might rise through cracks and form shallow reservoirs that future spacecraft could more easily study。 New simulations suggest that journey is unlikely because turbulent
A person’s sleeping brain may reveal warning signs of dementia long before memory problems begin。 Researchers used machine learning to analyze EEG recordings from about 7,000 adults and found that an older-than-expected “brain age” was tied to a sharply higher dementia risk。 Every additional 10 years of brain aging raised that risk by nearly 40%
NASA’s Swift Observatory observed a supermassive black hole ripping apart a star more than 30,000 light-years from the center of a distant galaxy。 The extraordinary flare briefly outshone its entire host galaxy in ultraviolet light and revealed a black hole about a million times the Sun’s mass
A quantum problem once described as impossible for classical computers has now been solved using relatively modest hardware。 Researchers used tensor networks to compress the overwhelming wave function created by hundreds of entangled qubits, allowing some calculations to run on a laptop。 Their results matched both theoretical predictions and simula