We consider the patch problem for the (Formula presented.) -(surface quasi-geostrophic) SQG system with the values (Formula presented.) and (Formula presented.) being the 2D Euler and the SQG equations respectively. It is well-known that the Euler patches are globally wellposed in non-endpoint… read more about this publication »
A canonical minimal free resolution of an arbitrary co-artinian lattice ideal over the polynomial ring is constructed over any field whose characteristic is 0 or any but finitely many positive primes. The differential has a closed-form combinatorial description as a sum over lattice paths in Zn of… read more about this publication »
The symmetric spaces that appear as moduli spaces in string theory and supergravity can be decomposed with explicit metrics using parabolic subgroups. The resulting isometry between the original moduli space and this decomposition can be used to find parametrizations of the moduli. One application… read more about this publication »
We study the relation between the coregularity, the index of log Calabi-Yau pairs and the complements of Fano varieties. We show that the index of a log Calabi-Yau pair of coregularity is at most, where is the Weil index of. This extends a recent result due to Filipazzi, Mauri and Moraga. We prove… read more about this publication »
The Homomorphism Preservation Theorem (HPT) of classical model theory states that a first-order sentence is preserved under homomorphisms if, and only if, it is equivalent to an existential-positive sentence. This theorem remains valid when restricted to finite structures, as demonstrated by the… read more about this publication »
We consider steady states of the two-dimensional incompressible Euler equations on T2 and construct smooth and singular steady states around a particular singular steady state. More precisely, we construct families of smooth and singular steady solutions that converge to the Bahouri–Chemin patch. read more about this publication »
We investigate a kinetic model for interacting particles whose masses are integer multiples of an elementary mass. These particles undergo binary collisions which preserve momentum and energy but during which some number of elementary masses can be exchanged between the particles. We derive a… read more about this publication »
Single-cell RNA sequencing has been widely used to investigate cell state transitions and gene dynamics of biological processes. Current strategies to infer the sequential dynamics of genes in a process typically rely on constructing cell pseudotime through cell trajectory inference. However, the… read more about this publication »
In this work, we seek to simulate rare transitions between metastable states using score-based generative models. An efficient method for generating high-quality transition paths is valuable for the study of molecular systems since data is often difficult to obtain. We develop two novel methods for… read more about this publication »
We study a family of structure-preserving deterministic numerical schemes for Lindblad equations. This family of schemes has a simple form and can systemically achieve arbitrary high-order accuracy in theory. Moreover, these schemes can also overcome the non-physical issues that arise from many… read more about this publication »
When auditing a redistricting plan, a persuasive method is to compare the plan with an ensemble of neutrally drawn redistricting plans. Ensembles are generated via algorithms that sample distributions on balanced graph partitions. To audit the partisan difference between the ensemble and a given… read more about this publication »
We show that the Euclidean 3-space R3 is stable for the Positive Mass Theorem in the following sense. Let (Mi,gi) be a sequence of complete asymptotically flat 3-manifolds with nonnegative scalar curvature and suppose that the ADM mass m(gi) of one end of Mi converges to 0. Then for all i, there is… read more about this publication »
We study Betti numbers of sequences of Riemannian manifolds which Benjamini–Schramm converge to their universal covers. Using the Price inequalities we developed elsewhere, we derive two distinct convergence results. First, under a negative Ricci curvature assumption and no assumption on sign of… read more about this publication »
Known as the no fast-forwarding theorem in quantum computing (see, e.g., Theorem 3 in [D. W. Berry et al., Comm. Math. Phys., 270 (2007), pp. 359-371], Theorem 5 in [A. M. Childs, Comm. Math. Phys., 294 (2010), pp. 581-603], and [R. Kothari, Efficient Simulation of Hamiltonians, M.S. thesis,… read more about this publication »
We study the index of log Calabi–Yau pairs (X, B) of coregularity 0. We show that 2λ(KX + B) ∼ 0, where λ is the Weil index of (X, B). This is in contrast to the case of klt Calabi–Yau varieties, where the index can grow doubly exponentially with the dimension. Our sharp bound on the index extends… read more about this publication »
We prove that a geometrically integral smooth projective 3-fold (Formula presented.) with nef anti-canonical class and negative Kodaira dimension over a finite field (Formula presented.) of characteristic (Formula presented.) and cardinality (Formula presented.) has a rational point. Additionally,… read more about this publication »
We show that elliptic Calabi–Yau threefolds form a bounded family. We also show that the same result holds for minimal terminal threefolds of Kodaira dimension 2, upon fixing the rate of growth of pluricanonical forms and the degree of a multisection of the Iitaka fibration. Both of these… read more about this publication »
In this note we describe basic geometric properties of p-harmonic forms and p-coclosed forms and use them to reprove vanishing theorems of Pansu and new injectivity theorems for the Lp-cohomology of simply connected, pinched negatively curved manifolds. We also provide a partial resolution of a… read more about this publication »
We consider nonnegative solutions of the quasilinear heat equation in one dimension. Our solutions may vanish and may be unbounded. The equation is then degenerate, and weak solutions are generally nonunique. We introduce a notion of strong solution that ensures uniqueness. For suitable initial… read more about this publication »
When studying out-of-equilibrium systems, one often excites the dynamics in some degrees of freedom while removing the excitation in others through damping. In order for the system to converge to a statistical steady state, the dynamics must transfer the energy from the excited modes to the… read more about this publication »
We study the period map of configurations of n points on the projective line constructed via a cyclic cover branching along these points. By considering the decomposition of its Hodge structure into eigenspaces, we establish the codimension of the locus where the eigenperiod map is still pure.… read more about this publication »
Numerical semigroups with multiplicity m are parametrized by integer points in a polyhedral cone Cm, according to Kunz. For the toric ideal of any such semigroup, the main result here constructs a free resolution whose overall structure is identical for all semigroups parametrized by the relative… read more about this publication »
Sampling a probability distribution with known likelihood is a fundamental task in computational science and engineering. Aiming at multimodality, we propose a new sampling method that takes advantage of both the birth-death process and exploration component. The main idea of this method is look… read more about this publication »
We show that if a composite θ-curve has (proper rational) unknotting number one, then it is the order 2 sum of a (proper rational) unknotting number one knot and a trivial θ-curve. We also prove similar results for 2-strand tangles and knotoids read more about this publication »