In [D. Rees, Hilbert functions and pseudo-rational local rings of dimension two, J. London Math. Soc. (2) 24 (1981) 467-479], Rees gave a characterization for the normal joint reduction number zero of two m-primary ideals in an analytically unramified Cohen-Macaulay local ring of dimension two. Rees' result is a generalization of Zariski's product theorem for complete ideals in a regular local ring of dimension two. The aim of this paper is to extend Rees' theorem for the ordinary powers of m-primary ideals II and JJ in a Cohen-Macaulay local ring of dimension two. Following Rees' approach, we define the modified Koszul homology modules M1r,s(ak,bk)Mr,s1(ak,bk) for a joint reduction (a,b)(a,b) of II and JJ. Under the additional assumption that the associated graded rings of II and JJ have positive depth, we obtain a characterization of the joint reduction number zero of II and JJ in terms of the vanishing of the module M10,0(a,b)M0,01(a,b), as well as in terms of the Hilbert coefficients and the bigraded Hilbert coefficients. More generally, we introduce the joint reduction lattice and study the vanishing of M1r,s(a,b)Mr,s1(a,b) for any r,s >= 0r,s >= 0. This gives a characterization for a vector (r,s)(r,s) to be in the joint reduction lattice of II and JJ. We also give a cohomological interpretation of these theorems by investigating the local cohomology modules of the bigraded extended Rees algebra. This gives another characterization for a vector (r,s)(r,s) to be in the joint reduction lattice and also extends a recent result of Masuti and Verma in [Local cohomology of bigraded Rees algebras and normal Hilbert coefficients, J. Pure Appl. Algebra218(5) (2014) 904-918] for ordinary powers of ideals
Given a choice of an ordered, orthonormal basis for a D-dimensional Hilbert space, one can define a discrete version of the Wigner function - a quasi-probability distribution which represents any quantum state as a real, normalized function on a discrete phase space. The Wigner function, in general, takes on negative values, and the amount of negativity in the Wigner function gives an operationally meaningful measure of the complexity of simulating the quantum state on a classical computer. Further, Wigner negativity also gives a lower bound on an entropic measure of spread complexity. In this paper, we study the growth of Wigner negativity for a generic initial state under time evolution with chaotic Hamiltonians. In [1], a perturbative argument was given to show that the Krylov basis minimizes the early time growth of Wigner negativity in the large-D limit. Using tools from random matrix theory, here we show that for a generic choice of basis, the Wigner negativity for a classical initial state becomes exponentially large in an O(1) amount of time evolution. On the other hand, we show that in the Krylov basis the negativity grows at most as a power law, and becomes exponentially large only at exponential times. We take this as evidence that the Krylov basis is ideally suited for a dual, semi-classical effective description of chaotic quantum dynamics for large-D at sub-exponential times. For the Gaussian unitary ensemble, this effective description is the q -> 0 limit of q-deformed JT gravity.
We study fair division of indivisible mixed manna (items whose values may be positive, negative, or zero) among agents with additive valuations. In this context, we establish that fairness—in terms of a relaxation of envy-freeness—and Pareto efficiency can always be achieved together. Specifically, our fairness guarantees are in terms of envy-freeness up to k reallocations ( EFR-k ): An allocation 𝒜 of the indivisible items is said to be EFR-k if there exists a subset R of at most k items such that, for each agent i, we can reassign items from within R (in 𝒜 ) and obtain an allocation, 𝒜^i , which is envy-free for i. We establish that, when allocating mixed manna among n agents with additive valuations, an EFR-(n-1) and Pareto optimal ( PO ) allocation 𝒜 always exists. Further, the individual envy-free allocations 𝒜^i , induced by reassignments, are also PO . In addition, we prove that such fair and efficient allocations are efficiently computable when the number of agents, n, is fixed. We also obtain positive results focusing on EFR by itself (and without the PO desideratum). Specifically, we show that an EFR-(n-1) allocation of mixed manna can be computed in polynomial time. In addition, we prove that when all the items are goods, an EFR-⌊ n/2 ⌋ allocation exists and can be computed efficiently. Here, the (n-1) bound is tight for chores and ⌊ n/2 ⌋ is tight for goods. Our results advance the understanding of fair and efficient allocation of indivisible mixed manna and rely on a novel application of the Knaster-Kuratowski-Mazurkiewicz (KKM) Theorem in discrete fair division. We utilize weighted welfare maximization, with perturbed valuations, to achieve Pareto efficiency, and overall, our techniques are notably different from existing market-based approaches.
Abstract We show that the near horizon of a generic non-extremal black hole (BH) can be understood in terms of a Carrollian geometry with two null directions, also called a String-Carroll (SC) geometry. The base space of this fibre-bundle structure is a sphere (or a plane for a black brane) and the fibre is the two-dimensional Rindler spacetime. We launch a detailed study of probes in this geometry. We study particle geodesics and scalar fields. The first part of the paper constructs geodesics and probe scalar fields directly in the SC geometry. We then look at a wide class of examples, including the Schwarzschild BH in asymptotically flat spacetimes, the black brane in AdS spacetimes, and Lifshitz black holes and construct the explicit maps to the SC geometry to obtain results specific to each case. These results are reproduced by considering the probe particles and fields in the original BH background and taking the near-horizon limit of the solutions. Our encyclopedia of examples establishes the notion of SC geometries as near-horizon geometries of non-extremal black objects, paving the way for a detailed, intricate future analysis of the quantum aspects of this geometry.
KMOC (Kosower, Maybee, and O’Connell) formalism is an approach to analyze classical scattering in gauge theories and gravity using a class of “inclusive” observables which can be computed solely from on-shell amplitudes [JHEP 02 (2019) 137]. This formalism has led to striking developments in the context of perturbative scattering, which corresponds to large impact parameter scattering. As a result, in its current form, the KMOC formulae cannot be directly applied to processes for generic values of the impact parameter. However, there is a domain where the relationship between classical radiation and on-shell amplitudes can be stretched beyond large impact parameter scattering. This regime is defined by the soft expansion of outgoing radiation. It is thus natural to ask whether such soft radiative fields can be computed using the basic paradigm set by the KMOC formalism. In this short note, we show that this is indeed the case for electromagnetic memory. In particular, we compute an inclusive observable associated with soft flux at ℐ^+ and show that, irrespective of the details of the hard scattering, this observable defines a non-perturbative formula for the electromagnetic memory in the classical limit. We argue that the result obtained for electromagnetic memory using the KMOC paradigm is consistent with that of [JHEP 09 (2018) 105], where the classical limit of the quantum soft theorem was derived using saddle-point analysis. The gravitational case, however, is qualitatively different due to the presence of the nonlinear memory effect, which requires knowledge of the hard amplitude. Consequently, unlike the electromagnetic memory, we have not been able to show consistency of the leading soft graviton theorem and the soft inclusive gravitational flux obtained using the KMOC formalism.
