scholarly journals Unexpected surfaces singular on lines in $${\mathbb {P}}^{3}$$

2020 ◽  
Author(s):  
Marcin Dumnicki ◽  
Brian Harbourne ◽  
Joaquim Roé ◽  
Tomasz Szemberg ◽  
Halszka Tutaj-Gasińska
Keyword(s):  
Linear Systems ◽  
Recent Work ◽  
Upper Bound ◽  
Open Problem ◽  
Waldschmidt Constants

AbstractWe study linear systems of surfaces in $${\mathbb {P}}^3$$ P 3 singular along general lines. Our purpose is to identify and classify special systems of such surfaces, i.e., those non-empty systems where the conditions imposed by the multiple lines are not independent. We prove the existence of four surfaces arising as (projective) linear systems with a single reduced member. Till now no such examples have been known. These are unexpected surfaces in the sense of recent work of Cook II, Harbourne, Migliore, and Nagel. It is an open problem if our list is complete, i.e., if it contains all reduced and irreducible unexpected surfaces based on lines in $${\mathbb {P}}^3$$ P 3 . As an application we find Waldschmidt constants of six general lines in $${\mathbb {P}}^3$$ P 3 and an upper bound for this invariant for seven general lines.

scholarly journals Polynomial Mixing of the Edge-Flip Markov Chain for Unbiased Dyadic Tilings

2018 ◽  
Vol 28 (3) ◽  
pp. 365-387
Author(s):  
S. CANNON ◽  
D. A. LEVIN ◽  
A. STAUFFER
Keyword(s):  
Markov Chain ◽  
Relaxation Time ◽  
Lower Bound ◽  
Upper Bound ◽  
Open Problem ◽  
Mixing Time ◽  
Perpendicular Bisector ◽  
Unit Square

We give the first polynomial upper bound on the mixing time of the edge-flip Markov chain for unbiased dyadic tilings, resolving an open problem originally posed by Janson, Randall and Spencer in 2002 [14]. A dyadic tiling of size n is a tiling of the unit square by n non-overlapping dyadic rectangles, each of area 1/n, where a dyadic rectangle is any rectangle that can be written in the form [a2−s, (a + 1)2−s] × [b2−t, (b + 1)2−t] for a, b, s, t ∈ ℤ⩾ 0. The edge-flip Markov chain selects a random edge of the tiling and replaces it with its perpendicular bisector if doing so yields a valid dyadic tiling. Specifically, we show that the relaxation time of the edge-flip Markov chain for dyadic tilings is at most O(n4.09), which implies that the mixing time is at most O(n5.09). We complement this by showing that the relaxation time is at least Ω(n1.38), improving upon the previously best lower bound of Ω(n log n) coming from the diameter of the chain.

scholarly journals Communication Complexity And Linearly Ordered Sets

2015 ◽  
Vol 29 (1) ◽  
pp. 93-117
Author(s):  
Mieczysław Kula ◽  
Małgorzata Serwecińska
Keyword(s):  
Upper Bound ◽  
Open Problem ◽  
Numerical Experiments ◽  
Communication Complexity ◽  
Ordered Sets ◽  
Finite Sets ◽  
The Family ◽  
Linear Lattices

AbstractThe paper is devoted to the communication complexity of lattice operations in linearly ordered finite sets. All well known techniques ([4, Chapter 1]) to determine the communication complexity of the infimum function in linear lattices disappoint, because a gap between the lower and upper bound is equal to O(log2n), where n is the cardinality of the lattice. Therefore our aim will be to investigate the communication complexity of the function more carefully. We consider a family of so called interval protocols and we construct the interval protocols for the infimum. We prove that the constructed protocols are optimal in the family of interval protocols. It is still open problem to compute the communication complexity of constructed protocols but the numerical experiments show that their complexity is less than the complexity of known protocols for the infimum function.

Robust State Estimation for Linear Systems

1993 ◽  
Vol 115 (1) ◽  
pp. 193-196
Author(s):  
S. S. Garimella ◽  
K. Srinivasan
Keyword(s):  
Dynamic System ◽  
State Estimation ◽  
Real Time ◽  
Linear Systems ◽  
Upper Bound ◽  
Estimation Error ◽  
System Model ◽  
Linear Dynamic ◽  
Modeling Errors ◽  
System States

Real-time state estimation of a linear dynamic system using an observer, in the presence of modeling errors in the system model used by the observer and uncertainty in the initial system states, is considered here. A guideline for designing observers for multioutput systems is established, based on an expression for an upper bound on the norm of the state estimation error derived in this paper. An example is presented to illustrate the usefulness of this guideline.

H∞ Robust State Feedback Sampled-Data Control of Interval Linear Systems

2020 ◽  
Author(s):  
Rafael M. Alves ◽  
André R. Fioravanti ◽  
Matheus Souza
Keyword(s):  
Control Problem ◽  
Linear Systems ◽  
Hybrid Systems ◽  
Upper Bound ◽  
State Feedback ◽  
Data Systems ◽  
Sampled Data ◽  
Data Control ◽  
Interval Linear ◽  
Interval Linear Systems

In this paper, we address the H∞ control problem for uncertain sampled-data systems rewritten as hybrid systems. The conditions proposed are formulated as intervals to ensure stability and design controllers that guarantee an upper bound for an associated H∞ norm. A numerical example points out the main features of the proposed method.

scholarly journals The Exact Security of PMAC with Two Powering-Up Masks

2019 ◽  
pp. 125-145 ◽  
Author(s):  
Yusuke Naito
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Open Problem ◽  
Random Function ◽  
Block Cipher ◽  
Random Permutation ◽  
Authentication Code ◽  
Open Problems ◽  
Main Research ◽  
Main Research Topic

PMAC is a rate-1, parallelizable, block-cipher-based message authentication code (MAC), proposed by Black and Rogaway (EUROCRYPT 2002). Improving the security bound is a main research topic for PMAC. In particular, showing a tight bound is the primary goal of the research, since Luykx et al.’s paper (EUROCRYPT 2016). Regarding the pseudo-random-function (PRF) security of PMAC, a collision of the hash function, or the difference between a random permutation and a random function offers the lower bound Ω(q2/2n) for q queries and the block cipher size n. Regarding the MAC security (unforgeability), a hash collision for MAC queries, or guessing a tag offers the lower bound Ω(q2m /2n + qv/2n) for qm MAC queries and qv verification queries (forgery attempts). The tight upper bound of the PRF-security O(q2/2n) of PMAC was given by Gaži et el. (ToSC 2017, Issue 1), but their proof requires a 4-wise independent masking scheme that uses 4 n-bit random values. Open problems from their work are: (1) find a masking scheme with three or less random values with which PMAC has the tight upper bound for PRF-security; (2) find a masking scheme with which PMAC has the tight upper bound for MAC-security.In this paper, we consider PMAC with two powering-up masks that uses two random values for the masking scheme. Using the structure of the powering-up masking scheme, we show that the PMAC has the tight upper bound O(q2/2n) for PRF-security, which answers the open problem (1), and the tight upper bound O(q2m /2n + qv/2n) for MAC-security, which answers the open problem (2). Note that these results deal with two-key PMACs, thus showing tight upper bounds of PMACs with single-key and/or with one powering-up mask are open problems.

ESTIMATE OF THE HAUSDORFF MEASURE OF THE SIERPINSKI TRIANGLE

Fractals ◽  
2009 ◽  
Vol 17 (02) ◽  
pp. 137-148
Author(s):  
PÉTER MÓRA
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Hausdorff Measure ◽  
Upper Bound ◽  
Open Problem ◽  
Dimensional Hausdorff Measure ◽  
Sierpinski Triangle ◽  
The One

It is well-known that the Hausdorff dimension of the Sierpinski triangle Λ is s = log 3/ log 2. However, it is a long standing open problem to compute the s-dimensional Hausdorff measure of Λ denoted by [Formula: see text]. In the literature the best existing estimate is [Formula: see text] In this paper we improve significantly the lower bound. We also give an upper bound which is weaker than the one above but everybody can check it easily. Namely, we prove that [Formula: see text] holds.

A note about a partial no-go theorem for quantum PCP

2011 ◽  
Vol 11 (11&12) ◽  
pp. 1019-1027
Author(s):  
Itai Itai Arad
Keyword(s):  
Critical Point ◽  
Upper Bound ◽  
Open Problem ◽  
Preliminary Step ◽  
Important Open Problem

This is not a disproof of the quantum PCP conjecture! In this note we use perturbation on the commuting Hamiltonian problem on a graph, based on results by Bravyi and Vyalyi, to provide a very partial no-go theorem for quantum PCP. Specifically, we derive an upper bound on how large the promise gap can be for the quantum PCP still to hold, as a function of the non-commuteness of the system. As the system becomes more and more commuting, the maximal promise gap shrinks. We view these results as possibly a preliminary step towards disproving the quantum PCP conjecture posed in \cite{ref:Aha09}. A different way to view these results is actually as indications that a critical point exists, beyond which quantum PCP indeed holds; in any case, we hope that these results will lead to progress on this important open problem.

scholarly journals Sur les solutions friables de l'équation a+b=c

2013 ◽  
Vol 154 (3) ◽  
pp. 439-463 ◽  
Author(s):  
SARY DRAPPEAU
Keyword(s):  
Lower Bound ◽  
Asymptotic Behaviour ◽  
Recent Work ◽  
Upper Bound ◽  
Smooth Function ◽  
Riemann Hypothesis ◽  
Smooth Solutions ◽  
Compactly Supported ◽  
The One

AbstractIn a recent paper [5], Lagarias and Soundararajan study the y-smooth solutions to the equation a+b=c. Conditionally under the Generalised Riemann Hypothesis, they obtain an estimate for the number of those solutions weighted by a compactly supported smooth function, as well as a lower bound for the number of bounded unweighted solutions. In this paper, we prove a more precise conditional estimate for the number of weighted solutions that is valid when y is relatively large with respect to x, so as to connect our estimate with the one obtained by La Bretèche and Granville in a recent work [2]. We also prove, conditionally under the Generalised Riemann Hypothesis, the conjectured upper bound for the number of bounded unweighted solutions, thus obtaining its exact asymptotic behaviour.

scholarly journals Stability of Interval Positive Fractional Discrete–Time Linear Systems

2018 ◽  
Vol 28 (3) ◽  
pp. 451-456 ◽  
Author(s):  
Tadeusz Kaczorek
Keyword(s):  
Linear Systems ◽  
Discrete Time ◽  
Upper Bound ◽  
Asymptotically Stable ◽  
Kharitonov Theorem ◽  
Time Linear

Abstract The aim of this work is to show that interval positive fractional discrete-time linear systems are asymptotically stable if and only if the respective lower and upper bound systems are asymptotically stable. The classical Kharitonov theorem is extended to interval positive fractional linear systems.

scholarly journals A Restricted Version of Wythoff's Game

10.37236/694 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Wen An Liu ◽  
Haifeng Li ◽  
Bei Li
Keyword(s):  
Upper Bound ◽  
Open Problem ◽  
Restricted Version

E. Duchêne and S. Gravier present the following open problem: In Wythoff's game, each player can either remove at most $R$ tokens from a single heap (i.e. there is an upper bound $R$ on the number of removing tokens), or remove the same number of tokens from both heaps but there is no upper bound on the number of removing tokens. This open problem is investigated and all its P-positions are given.

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