An Exponential Separation Between MA and AM Proofs of Proximity

Interactive proofs of proximity allow a sublinear-time verifier to check that a given input is close to the language, using a small amount of communication with a powerful (but untrusted) prover. In this work, we consider two natural minimally interactive variants of such proofs systems, in which the prover only sends a single message, referred to as the proof. The first variant, known as -proofs of Proximity (), is fully non-interactive, meaning that the proof is a function of the input only. The second variant, known as -proofs of Proximity (), allows the proof to additionally depend on the verifier's (entire) random string. The complexity of both s and s is the total number of bits that the verifier observes—namely, the sum of the proof length and query complexity. Our main result is an exponential separation between the power of s and s. Specifically, we exhibit an explicit and natural property $$\Pi$$ Π that admits an with complexity $$O(\log n)$$ O ( log n ) , whereas any for $$\Pi$$ Π has complexity $$\tilde{\Omega}(n^{1/4})$$ Ω ~ ( n 1 / 4 ) , where n denotes the length of the input in bits. Our lower bound also yields an alternate proof, which is more general and arguably much simpler, for a recent result of Fischer et al. (ITCS, 2014). Also, Aaronson (Quantum Information & Computation 2012) has shown a $$\Omega(n^{1/6})$$ Ω ( n 1 / 6 ) lower bound for the same property $$\Pi$$ Π .Lastly, we also consider the notion of oblivious proofs of proximity, in which the verifier's queries are oblivious to the proof. In this setting, we show that s can only be quadratically stronger than s. As an application of this result, we show an exponential separation between the power of public and private coin for oblivious interactive proofs of proximity.

Applications of a Grassmannian technique to hyperbolicity, Chow equivalency, and Seshadri constants

In this paper we further develop a Grassmannian technique used to prove results about very general hypersurfaces and present three applications. First, we provide a short proof of the Kobayashi conjecture given previously established results on the Green–Griffiths–Lang conjecture. Second, we completely resolve a conjecture of Chen, Lewis, and Sheng on the dimension of the space of Chow-equivalent points on a very general hypersurface, proving the remaining cases and providing a short, alternate proof for many of the previously known cases. Finally, we relate Seshadri constants of very general points to Seshadri constants of arbitrary points of very general hypersurfaces.

An application of a C2-estimate for a complex Monge–Ampère equation

By studying a complex Monge–Ampère equation, we present an alternate proof to a recent result of Chu-Lee-Tam concerning the projectivity of a compact Kähler manifold [Formula: see text] with [Formula: see text] for some integer [Formula: see text] with [Formula: see text], and the ampleness of the canonical line bundle [Formula: see text].

Another Proof of e^{x/y} being irrational

Continued fractions are used to give an alternate proof of e^{x/y} is irrational.

Min-Max Theory for Cell Complexes

In the study of smooth functions on manifolds, min-max theory provides a mechanism for identifying critical values of a function. We introduce a discretized version of this theory associated to a discrete Morse function on a (regular) cell complex. As applications we prove a discrete version of the mountain pass lemma and give an alternate proof of a discrete Lusternik–Schnirelmann theorem.

The Absolute Orders on the Coxeter Groups $A_n$ and $B_n$ are Sperner

There are several classes of ranked posets related to reflection groups which are known to have the Sperner property, including the Bruhat orders and the generalized noncrossing partition lattices (i.e., the maximal intervals in absolute orders). In 2019, Harper–Kim proved that the absolute orders on the symmetric groups are (strongly) Sperner. In this paper, we give an alternate proof that extends to the signed symmetric groups and the dihedral groups. Our simple proof uses techniques inspired by Ford–Fulkerson's theory of networks and flows, and a product theorem.

Orbit design and thruster requirement for various constant arm space mission concepts for gravitational-wave observation

In previous papers, we have addressed the issues of orbit design and thruster requirement for the constant arm versions of Astrodynamical Middle-frequency Interferometric Gravitational-wave Observatory (AMIGO) mission concept and for the constant arm gravitational wave (GW) mission concept of Atom Interferometric Gravitational-wave Space Observatory (AIGSO). In this paper, we apply similar methods to the orbit design and thruster requirement for the constant arm GW missions B-DECIGO and DECIGO, and estimate the yearly propellant requirements at the specific impulse [Formula: see text][Formula: see text]s and [Formula: see text][Formula: see text]s. For the geocentric orbit options of B-DECIGO which we have explored, the fuel mass requirement is a concern. For the heliocentric orbit options of B-DECIGO and DECIGO, the fuel requirement to keep the arm equal and constant should be easily satisfied. Furthermore, we explore the thruster and propellant requirements for constant arm versions of LISA and TAIJI missions and find the fuel mass requirement is not a show stopper either. The proof mass actuation noise is a concern. To have enough dynamical range, an alternate proof mass is required. Detailed laboratory study is warranted.