Multiple seeds sensitivity using a single seed with threshold

2015 ◽  
Vol 13 (04) ◽  
pp. 1550011 ◽  
Author(s):  
Lavinia Egidi ◽  
Giovanni Manzini
Keyword(s):  
Prime Number ◽  
Similarity Search ◽  
A Posteriori ◽  
Quadratic Residues ◽  
Spaced Seeds ◽  
Trade Offs ◽  
Sensitivity And Selectivity ◽  
Practical Limit ◽  
Single Seeds ◽  
Number Of Seeds

Spaced seeds are a fundamental tool for similarity search in biosequences. The best sensitivity/selectivity trade-offs are obtained using many seeds simultaneously: This is known as the multiple seed approach. Unfortunately, spaced seeds use a large amount of memory and the available RAM is a practical limit to the number of seeds one can use simultaneously. Inspired by some recent results on lossless seeds, we revisit the approach of using a single spaced seed and considering two regions homologous if the seed hits in at least t sufficiently close positions. We show that by choosing the locations of the don't care symbols in the seed using quadratic residues modulo a prime number, we derive single seeds that when used with a threshold t > 1 have competitive sensitivity/selectivity trade-offs, indeed close to the best multiple seeds known in the literature. In addition, the choice of the threshold t can be adjusted to modify sensitivity and selectivity a posteriori, thus enabling a more accurate search in the specific instance at issue. The seeds we propose also exhibit robustness and allow flexibility in usage.

INFINITE-DIMENSIONAL GENERALIZED CONTINUED FRACTIONS, DISTRIBUTION OF QUADRATIC RESIDUES AND NON-RESIDUES, AND ERGODIC THEORY

2002 ◽  
Vol 05 (04) ◽  
pp. 555-570
Author(s):  
L. D. PUSTYL'NIKOV
Keyword(s):  
Prime Number ◽  
Ergodic Theory ◽  
Continued Fractions ◽  
Classical Problem ◽  
Infinite Dimensional ◽  
Ergodic Properties ◽  
Quadratic Residues ◽  
Generalized Continued Fractions

A new theory of generalized continued fractions for infinite-dimensional vectors with integer components is constructed. The results of this theory are applied to the classical problem on the distribution of quadratic residues and non-residues modulo a prime number and are based on the study of ergodic properties of some infinite-dimensional transformations.

scholarly journals Sensitivity-Selectivity Trade-Offs in Surface Ionization Gas Detection

Nanomaterials ◽  
2018 ◽  
Vol 8 (12) ◽  
pp. 1017 ◽  
Author(s):  
Gerhard Müller ◽  
J. Prades ◽  
Angelika Hackner ◽  
Andrea Ponzoni ◽  
Elisabetta Comini ◽  
...  
Keyword(s):  
Illicit Drugs ◽  
Surface Ionization ◽  
Micro Electro Mechanical Systems ◽  
Selective Method ◽  
Decay Products ◽  
Trade Offs ◽  
High Proton ◽  
Sensitivity And Selectivity ◽  
High Level ◽  
Chemical Selectivity

Surface ionization (SI) provides a simple, sensitive, and selective method for the detection of high-proton affinity substances, such as organic decay products, medical and illicit drugs as well as a range of other hazardous materials. Tests on different kinds of SI sensors showed that the sensitivity and selectivity of such devices is not only dependent on the stoichiometry and nanomorphology of the emitter materials, but also on the shape of the electrode configurations that are used to read out the SI signals. Whereas, in parallel-plate capacitor devices, different kinds of emitter materials exhibit a high level of amine-selectivity, MEMS (micro-electro-mechanical-systems) and NEMS (nanowire) versions of SI sensors employing the same kinds of emitter materials provide significantly higher sensitivity, however, at the expense of a reduced chemical selectivity. In this paper, it is argued that such sensitivity-selectivity trade-offs arise from unselective physical ionization phenomena that occur in the high-field regions immediately adjacent to the surfaces of sharply curved MEMS (NEMS) emitter and collector electrodes.

scholarly journals A UNIFYING FRAMEWORK FOR SEED SENSITIVITY AND ITS APPLICATION TO SUBSET SEEDS

2006 ◽  
Vol 04 (02) ◽  
pp. 553-569 ◽  
Author(s):  
GREGORY KUCHEROV ◽  
LAURENT NOÉ ◽  
MIKHAIL ROYTBERG
Keyword(s):  
Probability Distribution ◽  
Similarity Search ◽  
Finite Automata ◽  
Experimental Results ◽  
Spaced Seeds ◽  
Sensitivity Problem ◽  
Three Components

We propose a general approach to compute the seed sensitivity, that can be applied to different definitions of seeds. It treats separately three components of the seed sensitivity problem — a set of target alignments, an associated probability distribution, and a seed model — that are specified by distinct finite automata. The approach is then applied to a new concept of subset seeds for which we propose an efficient automaton construction. Experimental results confirm that sensitive subset seeds can be efficiently designed using our approach, and can then be used in similarity search producing better results than ordinary spaced seeds.

scholarly journals On spaced seeds for similarity search

2004 ◽  
Vol 138 (3) ◽  
pp. 253-263 ◽  
Author(s):  
Uri Keich ◽  
Ming Li ◽  
Bin Ma ◽  
John Tromp
Keyword(s):  
Similarity Search ◽  
Spaced Seeds

scholarly journals Better spaced seeds using Quadratic Residues

2013 ◽  
Vol 79 (7) ◽  
pp. 1144-1155 ◽  
Author(s):  
Lavinia Egidi ◽  
Giovanni Manzini
Keyword(s):  
Quadratic Residues ◽  
Spaced Seeds

scholarly journals Counting certain quadratic partitions of zero modulo a prime number

2021 ◽  
Vol 19 (1) ◽  
pp. 198-211
Author(s):  
Wang Xiao ◽  
Aihua Li
Keyword(s):  
Number Theory ◽  
Prime Number ◽  
Diophantine Equation ◽  
Type I ◽  
Type Ii ◽  
Number Of Solutions ◽  
Quadratic Residues

Abstract Consider an odd prime number p ≡ 2 ( mod 3 ) p\equiv 2\hspace{0.3em}\left(\mathrm{mod}\hspace{0.3em}3) . In this paper, the number of certain type of partitions of zero in Z / p Z {\mathbb{Z}}\hspace{-0.1em}\text{/}\hspace{-0.1em}p{\mathbb{Z}} is calculated using a combination of elementary combinatorics and number theory. The focus is on the three-part partitions of 0 in Z / p Z {\mathbb{Z}}\hspace{-0.1em}\text{/}\hspace{-0.1em}p{\mathbb{Z}} with all three parts chosen from the set of non-zero quadratic residues mod p p . Such partitions are divided into two types. Those with exactly two of the three parts identical are classified as type I. The type II partitions are those with all three parts being distinct. The number of partitions of each type is given. The problem of counting such partitions is well related to that of counting the number of non-trivial solutions to the Diophantine equation x 2 + y 2 + z 2 = 0 {x}^{2}+{y}^{2}+{z}^{2}=0 in the ring Z / p Z {\mathbb{Z}}\hspace{-0.1em}\text{/}\hspace{-0.1em}p{\mathbb{Z}} . Correspondingly, solutions to this equation are also classified as type I or type II. We give the number of solutions to the equation corresponding to each type.

THE CONTINUING SEARCH FOR LARGE ELITE PRIMES

2009 ◽  
Vol 05 (02) ◽  
pp. 209-218 ◽  
Author(s):  
ALAIN CHAUMONT ◽  
JOHANNES LEICHT ◽  
TOM MÜLLER ◽  
ANDREAS REINHART
Keyword(s):  
Prime Number ◽  
Systematic Search ◽  
Open Problems ◽  
Fermat Numbers ◽  
Quadratic Residues ◽  
Modulo P

A prime number p is called elite if only finitely many Fermat numbers 22n + 1 are quadratic residues modulo p. So far, all 21 elite primes less than 250 billion were known, together with 24 larger items. We completed the systematic search up to the range of 2.5 · 1012 finding six more such numbers. Moreover, 42 new elites larger than this bound were found, among which the largest has 374 596 decimal digits. A survey on the knowledge about elite primes together with some open problems and conjectures are presented.

scholarly journals A computable formula for the class number of the imaginary quadratic field $ \mathbb Q(\sqrt{-p}), \ p = 4n-1 $

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jorge Garcia Villeda
Keyword(s):  
Prime Number ◽  
Class Number ◽  
Quadratic Field ◽  
Dirichlet Character ◽  
Imaginary Quadratic Field ◽  
Kronecker Symbol ◽  
Quadratic Residues ◽  
Elementary Methods ◽  
Computable Formula

<p style='text-indent:20px;'>Using elementary methods, we count the quadratic residues of a prime number of the form <inline-formula><tex-math id="M2">\begin{document}$ p = 4n-1 $\end{document}</tex-math></inline-formula> in a manner that has not been explored before. The simplicity of the pattern found leads to a novel formula for the class number <inline-formula><tex-math id="M3">\begin{document}$ h $\end{document}</tex-math></inline-formula> of the imaginary quadratic field <inline-formula><tex-math id="M4">\begin{document}$ \mathbb Q(\sqrt{-p}). $\end{document}</tex-math></inline-formula> Such formula is computable and does not rely on the Dirichlet character or the Kronecker symbol at all. Examples are provided and formulas for the sum of the quadratic residues are also found.</p>

Practical Correction of Three Fold Astigmatism in the Philips CM TEM

1996 ◽  
Vol 54 ◽  
pp. 418-419
Author(s):  
Arno J. Bleeker ◽  
Mark H.F. Overwijk ◽  
Max T. Otten
Keyword(s):  
Optical Properties ◽  
Phase Contrast ◽  
The Other ◽  
Objective Lens ◽  
Symmetric Part ◽  
A Posteriori ◽  
Contrast Transfer Function ◽  
High Brightness ◽  
Rotationally Symmetric ◽  
Brightness Field

With the improvement of the optical properties of the modern TEM objective lenses the point resolution is pushed beyond 0.2 nm. The objective lens of the CM300 UltraTwin combines a Cs of 0. 65 mm with a Cc of 1.4 mm. At 300 kV this results in a point resolution of 0.17 nm. Together with a high-brightness field-emission gun with an energy spread of 0.8 eV the information limit is pushed down to 0.1 nm. The rotationally symmetric part of the phase contrast transfer function (pctf), whose first zero at Scherzer focus determines the point resolution, is mainly determined by the Cs and defocus. Apart from the rotationally symmetric part there is also the non-rotationally symmetric part of the pctf. Here the main contributors are not only two-fold astigmatism and beam tilt but also three-fold astigmatism. The two-fold astigmatism together with the beam tilt can be corrected in a straight-forward way using the coma-free alignment and the objective stigmator. However, this only works well when the coefficient of three-fold astigmatism is negligible compared to the other aberration coefficients. Unfortunately this is not generally the case with the modern high-resolution objective lenses. Measurements done at a CM300 SuperTwin FEG showed a three fold-astigmatism of 1100 nm which is consistent with measurements done by others. A three-fold astigmatism of 1000 nm already sinificantly influences the image at a spatial frequency corresponding to 0.2 nm which is even above the point resolution of the objective lens. In principle it is possible to correct for the three-fold astigmatism a posteriori when through-focus series are taken or when off-axis holography is employed. This is, however not possible for single images. The only possibility is then to correct for the three-fold astigmatism in the microscope by the addition of a hexapole corrector near the objective lens.

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