scholarly journals Krein–Rutman type property and exponential separation of a noncompact operator

2019 ◽  
Vol 147 (11) ◽  
pp. 4771-4780
Author(s):  
Lirui Feng ◽  
Jianhong Wu
Keyword(s):  
Exponential Separation ◽  
Type Property

scholarly journals An Exponential Separation Between MA and AM Proofs of Proximity

2021 ◽  
Vol 30 (2) ◽  
Author(s):  
Tom Gur ◽  
Yang P. Liu ◽  
Ron D. Rothblum
Keyword(s):  
Quantum Information ◽  
Lower Bound ◽  
Recent Result ◽  
Query Complexity ◽  
Random String ◽  
Natural Property ◽  
Public And Private ◽  
Alternate Proof ◽  
Interactive Proofs ◽  
Exponential Separation

AbstractInteractive 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.

The randomized communication complexity of revenue maximization

2021 ◽  
Vol 19 (2) ◽  
pp. 75-83
Author(s):  
Aviad Rubinstein ◽  
Junyao Zhao
Keyword(s):  
Lower Bounds ◽  
Communication Complexity ◽  
Social Choice Rule ◽  
Valuation Function ◽  
Optimal Auctions ◽  
Incentive Compatible ◽  
First Approximation ◽  
Exponential Separation ◽  
Open Question

We study the communication complexity of incentive compatible auction-protocols between a monopolist seller and a single buyer with a combinatorial valuation function over n items [Rubinstein and Zhao 2021]. Motivated by the fact that revenue-optimal auctions are randomized [Thanassoulis 2004; Manelli and Vincent 2010; Briest et al. 2010; Pavlov 2011; Hart and Reny 2015] (as well as by an open problem of Babaioff, Gonczarowski, and Nisan [Babaioff et al. 2017]), we focus on the randomized communication complexity of this problem (in contrast to most prior work on deterministic communication). We design simple, incentive compatible, and revenue-optimal auction-protocols whose expected communication complexity is much (in fact infinitely) more efficient than their deterministic counterparts. We also give nearly matching lower bounds on the expected communication complexity of approximately-revenue-optimal auctions. These results follow from a simple characterization of incentive compatible auction-protocols that allows us to prove lower bounds against randomized auction-protocols. In particular, our lower bounds give the first approximation-resistant, exponential separation between communication complexity of incentivizing vs implementing a Bayesian incentive compatible social choice rule, settling an open question of Fadel and Segal [Fadel and Segal 2009].

scholarly journals Catastrophic Household Expenditure for Healthcare in Turkey: Clustering Analysis of Categorical Data

Data ◽  
2019 ◽  
Vol 4 (3) ◽  
pp. 112
Author(s):  
Onur Dogan ◽  
Gizem Kaya ◽  
Aycan Kaya ◽  
Hidayet Beyhan
Keyword(s):  
Health Expenditure ◽  
National Income ◽  
Health Spending ◽  
Health Problems ◽  
Catastrophic Health Expenditure ◽  
Catastrophic Expenditure ◽  
Household Level ◽  
Expenditure Rate ◽  
Type Property ◽  
Similarity And Difference

The amount of health expenditure at the household level is one of the most basic indicators of development in countries. In many countries, health expenditure increases relative to national income. If out-of-pocket health spending is higher than the income or too high, this indicates an economical alarm that causes a lower life standard, called catastrophic health expenditure. Catastrophic expenditure may be affected by many factors such as household type, property status, smoking and drinking alcohol habits, being active in sports, and having private health insurance. The study aims to investigate households with respect to catastrophic health expenditure by the clustering method. Clustering enables one to see the main similarity and difference between the groups. The results show that there are significant and interesting differences between the five groups. C4 households earn more but spend less money on health problems by the rate of 3.10% because people who do physical exercises regularly have fewer health problems. A household with a family with one adult, landlord and three people in total (mother or father and two children) in the cluster C5 earns much money and spends large amounts for health expenses than other clusters. C1 households with elementary families with three children, and who do not pay rent although they are not landlords have the highest catastrophic health expenditure. Households in C3 have a rate of 3.83% health expenditure rate on average, which is higher than other clusters. Households in the cluster C2 make the most catastrophic health expenditure.

scholarly journals Prime numbers with a certain extremal type property

2018 ◽  
Vol 17 (1) ◽  
pp. 127-151
Author(s):  
Edward Tutaj
Keyword(s):  
Riemann Hypothesis ◽  
Convex Set ◽  
Counting Function ◽  
Numerical Data ◽  
Prime Numbers ◽  
Affine Function ◽  
Piecewise Affine ◽  
Order Of Magnitude ◽  
Type Property ◽  
Prime Counting Function

Abstract The convex hull of the subgraph of the prime counting function x → π(x) is a convex set, bounded from above by a graph of some piecewise affine function x → (x). The vertices of this function form an infinite sequence of points $({e_k},\pi ({e_k}))_1^\infty $ . The elements of the sequence (ek)1∞ shall be called the extremal prime numbers. In this paper we present some observations about the sequence (ek)1∞ and we formulate a number of questions inspired by the numerical data. We prove also two – it seems – interesting results. First states that if the Riemann Hypothesis is true, then ${{{e_k} + 1} \over {{e_k}}} = 1$ . The second, also depending on Riemann Hypothesis, describes the order of magnitude of the differences between consecutive extremal prime numbers.

scholarly journals A Ramsey-type property in additive number theory

1985 ◽  
Vol 27 ◽  
pp. 5-10
Author(s):  
S. A. Burr ◽  
P. Erdös
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Additive Number Theory ◽  
Additive Number ◽  
Large Positive Integer ◽  
Ramsey Type ◽  
Type Property ◽  
Positive Integers

Let A be a sequence of positive integers. Define P(A) to be the set of all integers representable as a sum of distinct terms of A. Note that if A contains a repeated value, we are free to use it as many times as it occurs in A. We call A complete if every sufficiently large positive integer is in P(A), and entirely complete if every positive integer is in P(A). Completeness properties have received considerable, if somewhat sporadic, attention over the years. See Chapter 6 of [3] for a survey.

scholarly journals An Exponential Separation between Randomized and Deterministic Complexity in the LOCAL Model

2019 ◽  
Vol 48 (1) ◽  
pp. 122-143 ◽  
Author(s):  
Yi-Jun Chang ◽  
Tsvi Kopelowitz ◽  
Seth Pettie
Keyword(s):  
Local Model ◽  
Exponential Separation
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