scholarly journals An asymptotically optimal lower bound on the OBDD size of the middle bit of multiplication for the pairwise ascending variable order

2010 ◽  
Vol 158 (11) ◽  
pp. 1195-1204 ◽  
Author(s):  
Martin Sauerhoff
Keyword(s):  
Lower Bound ◽  
Variable Order ◽  
Asymptotically Optimal ◽  
Optimal Lower Bound

On the Inf-Sup Constant of a Triangular Spectral Method for the Stokes Equations

2016 ◽  
Vol 16 (3) ◽  
pp. 507-522 ◽  
Author(s):  
Yanhui Su ◽  
Lizhen Chen ◽  
Xianjuan Li ◽  
Chuanju Xu
Keyword(s):  
Lower Bound ◽  
Error Estimate ◽  
Spectral Method ◽  
Stokes Equations ◽  
Necessary Condition ◽  
Well Posedness ◽  
Saddle Point Problems ◽  
Numerical Examples ◽  
Lbb Condition ◽  
Optimal Lower Bound

AbstractThe Ladyženskaja–Babuška–Brezzi (LBB) condition is a necessary condition for the well-posedness of discrete saddle point problems stemming from discretizing the Stokes equations. In this paper, we prove the LBB condition and provide the (optimal) lower bound for this condition for the triangular spectral method proposed by L. Chen, J. Shen, and C. Xu in [3]. Then this lower bound is used to derive an error estimate for the pressure. Some numerical examples are provided to confirm the theoretical estimates.

scholarly journals A note on the number of irrational odd zeta values

2020 ◽  
Vol 156 (8) ◽  
pp. 1699-1717
Author(s):  
Li Lai ◽  
Pin Yu
Keyword(s):  
Optimal Design ◽  
Lower Bound ◽  
Closed Form ◽  
Rational Functions ◽  
Asymptotically Optimal ◽  
Zeta Values

AbstractWe prove that, for any small $\varepsilon > 0$, the number of irrationals among the following odd zeta values: $\zeta (3),\zeta (5),\zeta (7),\ldots ,\zeta (s)$ is at least $( c_0 - \varepsilon )({s^{1/2}}/{(\log s)^{1/2}})$, provided $s$ is a sufficiently large odd integer with respect to $\varepsilon$. The constant $c_0 = 1.192507\ldots$ can be expressed in closed form. Our work improves the lower bound $2^{(1-\varepsilon )({\log s}/{\log \log s})}$ of the previous work of Fischler, Sprang and Zudilin. We follow the same strategy of Fischler, Sprang and Zudilin. The main new ingredient is an asymptotically optimal design for the zeros of the auxiliary rational functions, which relates to the inverse totient problem.

An optimal lower bound for nonregular languages

1994 ◽  
Vol 52 (6) ◽  
pp. 339 ◽  
Author(s):  
A. Bertoni ◽  
Carlo Mereghetti ◽  
Giovanni Pighizzini
Keyword(s):  
Lower Bound ◽  
Optimal Lower Bound

scholarly journals Extreme Values of the Riemann Zeta Function on the 1-Line

2017 ◽  
Vol 2019 (22) ◽  
pp. 6924-6932 ◽  
Author(s):  
Christoph Aistleitner ◽  
Kamalakshya Mahatab ◽  
Marc Munsch
Keyword(s):  
Lower Bound ◽  
Extreme Values ◽  
Mean Square ◽  
Self Similarity ◽  
New Variant ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
The Mean ◽  
Resonator Method ◽  
Optimal Lower Bound

Abstract We prove that there are arbitrarily large values of t such that $|\zeta (1+it)| \geq e^{\gamma } (\log _{2} t +\log _{3} t) + \mathcal{O}(1)$. This essentially matches the prediction for the optimal lower bound in a conjecture of Granville and Soundararajan. Our proof uses a new variant of the “long resonator” method. While earlier implementations of this method crucially relied on a “sparsification” technique to control the mean-square of the resonator function, in the present paper we exploit certain self-similarity properties of a specially designed resonator function.

An optimal lower bound on the number of variables for graph identification

COMBINATORICA ◽  
1992 ◽  
Vol 12 (4) ◽  
pp. 389-410 ◽  
Author(s):  
Jin-Yi Cai ◽  
Martin F�rer ◽  
Neil Immerman
Keyword(s):  
Lower Bound ◽  
Optimal Lower Bound

scholarly journals Asymptotically optimal minimizers schemes

2018 ◽  
Author(s):  
Guillaume Marçais ◽  
Dan DeBlasio ◽  
Carl Kingsford
Keyword(s):  
Asymptotic Behavior ◽  
Lower Bound ◽  
Efficient Algorithm ◽  
Memory Usage ◽  
Theoretical Understanding ◽  
Asymptotically Optimal ◽  
Bioinformatics Tools ◽  
Bioinformatics Software ◽  
Run Time ◽  
Local Schemes

AbstractMotivationThe minimizers technique is a method to sample k-mers that is used in many bioinformatics software to reduce computation, memory usage and run time. The number of applications using minimizers keeps on growing steadily. Despite its many uses, the theoretical understanding of minimizers is still very limited. In many applications, selecting as few k-mers as possible (i.e. having a low density) is beneficial. The density is highly dependent on the choice of the order on the k-mers. Different applications use different orders, but none of these orders are optimal. A better understanding of minimizers schemes, and the related local and forward schemes, will allow designing schemes with lower density, and thereby making existing and future bioinformatics tools even more efficient.ResultsFrom the analysis of the asymptotic behavior of minimizers, forward and local schemes, we show that the previously believed lower bound on minimizers schemes does not hold, and that schemes with density lower than thought possible actually exist. The proof is constructive and leads to an efficient algorithm to compare k-mers. These orders are the first known orders that are asymptotically optimal. Additionally, we give improved bounds on the density achievable by the 3 type of [email protected]@cs.cmu.edu

Neural Nets with Superlinear VC-Dimension

1994 ◽  
Vol 6 (5) ◽  
pp. 877-884 ◽  
Author(s):  
Wolfgang Maass
Keyword(s):  
Lower Bound ◽  
Large Class ◽  
Neural Nets ◽  
New Method ◽  
Neural Net ◽  
Vc Dimension ◽  
Asymptotically Optimal ◽  
Linear Threshold

It has been known for quite a while that the Vapnik-Chervonenkis dimension (VC-dimension) of a feedforward neural net with linear threshold gates is at most O(w · log w), where w is the total number of weights in the neural net. We show in this paper that this bound is in fact asymptotically optimal. More precisely, we exhibit for any depth d ≥ 3 a large class of feedforward neural nets of depth d with w weights that have VC-dimension Ω(w · log w). This lower bound holds even if the inputs are restricted to Boolean values. The proof of this result relies on a new method that allows us to encode more “program-bits” in the weights of a neural net than previously thought possible.

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