Heights of Divisors of xn – 1

Integers ◽  
2011 ◽  
Vol 11 (4) ◽  
Author(s):  
Lola Thompson
Keyword(s):  
Upper Bound ◽  
Cyclotomic Polynomials ◽  
Absolute Value ◽  
Integer Coefficients ◽  
The Subject

AbstractThe height of a polynomial with integer coefficients is the largest coefficient in absolute value. Many papers have been written on the subject of bounding heights of cyclotomic polynomials. One result, due to H. Maier, gives a best possible upper bound of

scholarly journals Thue Inequalities With Few Coefficients

2020 ◽  
Author(s):  
Paloma Bengoechea
Keyword(s):  
Upper Bound ◽  
Binary Form ◽  
Number Of Solutions ◽  
Absolute Value ◽  
Integer Coefficients ◽  
The Absolute

Abstract Let $F(x, y)$ be a binary form with integer coefficients, degree $n\geq 3$, and irreducible over the rationals. Suppose that only $s + 1$ of the $n + 1$ coefficients of $F$ are nonzero. We show that the Thue inequality $|F(x,y)|\leq m$ has $\ll s m^{2/n}$ solutions provided that the absolute value of the discriminant $D(F)$ of $F$ is large enough. We also give a new upper bound for the number of solutions of $|F(x,y)|\leq m$, with no restriction on the discriminant of $F$ that depends mainly on $s$ and $m$, and slightly on $n$. Our bound becomes independent of $m$ when $m<|D(F)|^{2/(5(n-1))}$, and also independent of $n$ if $|D(F)|$ is large enough.

scholarly journals A Systematic Study of the Best Fitting Line Problem

2021 ◽  
Author(s):  
Giorgio Gambirasio
Keyword(s):  
Center Of Mass ◽  
Vertical Axis ◽  
Absolute Error ◽  
Optimization Method ◽  
Least Square ◽  
Absolute Value ◽  
Meeting Point ◽  
Best Fitting ◽  
The Subject ◽  
Comprehensive Study

AbstractThe classical approach for tackling the problem of drawing the 'best fitting line' through a plot of experimental points (here called a scenario) is the least square process applied to the errors along the vertical axis. However, more elaborate processes exist or may be found. In this report, we present a comprehensive study on the subject. Five possible processes are identified: two of them (respectively called VE, HE) measure errors along one axis, and the remaining three (respectively called YE, PE, and AE) take into consideration errors along both axes. Since the axes and their corresponding errors may have different physical dimensions, a procedure is proposed to compensate for this difference so that all processes could express their answers in the same consistent dimensions. As usual, to avoid mutual cancellation, errors are squared or taken in their absolute value. The two cases are separately studied.In the case of squared errors, the five processes are tested in many scenarios of experimental points, both analytically (using the software Mathematica) and numerically (with programs written on Python platform employing the Nelder-Mead optimization method). The investigation showed the possible existence of multiple solutions. Different answers originating from different starting points in Nelder?Mead correspond to solutions revealed by analytic search with Mathematica. For each scenario of experimental points, it was found that the best lines of the five processes intercept at a common point. Furthermore, the point of intercept happens to coincide with the 'center of mass' of the scenario. This fact is described by stating the existence of an empirical 'Meeting Point Law'. The case of absolute errors is only treated numerically, with Nelder?Mead minimizer. As expected, the absolute error option shows greater robustness against outliers than the square error option, for all processes. The Meeting Point Law is not valid in this case.By taking the value of minimized error as a criterion, the five processes are compared for efficiency. On average, processes PE and AE, that consider errors along both axes, resulted in the smallest minimized error and may be considered the best processes. Processes that rely on errors along a single axis (VE, HE) stay at the second place. In all cases, YE is the process that results in the largest minimized errors

VII.—On a Collection of Antiquities from the Early Iron Age Cemetery of Hallstatt, presented to the British Museum by Lord Avebury, 1916

Archaeologia ◽  
1916 ◽  
Vol 67 ◽  
pp. 145-162
Author(s):  
C. Hercules Read ◽  
Reginald A. Smith
Keyword(s):  
Iron Age ◽  
Turning Point ◽  
British Museum ◽  
Early Iron Age ◽  
Absolute Value ◽  
The Arts ◽  
History Of ◽  
The Subject ◽  
The Absolute

The important series of antiquities that forms the subject of this communication was discovered at Hallstatt in the Salzkammergut, Austria, about the year 1869. The exploration was undertaken at the instance of Sir John Lubbock (afterwards Lord Avebury), and it is believed that a journal was kept of the daily results, as appears to have been the case in all instances where authorized digging took place on the site. Unluckily in the interval between 1869 and the present time the journal referring to Lord Avebury's exploration has disappeared, and we thus lack an important part of the information that it should have furnished, viz. the indications as to what objects were associated together, and whether the interments to which they belonged were by cremation or by inhumation. While this loss is much to be regretted, yet the absolute value and importance of the series is still very great, both as typical of the period which stands prominent as the classical example of a cultural turning-point in the history of the arts, and as filling a very serious gap in the evolutionary series in the national collection.

scholarly journals On Graphs with Cyclic Defect or Excess

10.37236/415 ◽  
2010 ◽  
Vol 17 (1) ◽  
Author(s):  
Charles Delorme ◽  
Guillermo Pineda-Villavicencio
Keyword(s):  
Upper Bound ◽  
Adjacency Matrix ◽  
Minimum Degree ◽  
Integer Coefficients ◽  
Disjoint Cycles ◽  
The Matrix ◽  
Odd Degree ◽  
Unexplored Area ◽  
Trivial Graph ◽  
Vertex Disjoint

The Moore bound constitutes both an upper bound on the order of a graph of maximum degree $d$ and diameter $D=k$ and a lower bound on the order of a graph of minimum degree $d$ and odd girth $g=2k+1$. Graphs missing or exceeding the Moore bound by $\epsilon$ are called graphs with defect or excess $\epsilon$, respectively. While Moore graphs (graphs with $\epsilon=0$) and graphs with defect or excess 1 have been characterized almost completely, graphs with defect or excess 2 represent a wide unexplored area. Graphs with defect (excess) 2 satisfy the equation $G_{d,k}(A) = J_n + B$ ($G_{d,k}(A) = J_n - B$), where $A$ denotes the adjacency matrix of the graph in question, $n$ its order, $J_n$ the $n\times n$ matrix whose entries are all 1's, $B$ the adjacency matrix of a union of vertex-disjoint cycles, and $G_{d,k}(x)$ a polynomial with integer coefficients such that the matrix $G_{d,k}(A)$ gives the number of paths of length at most $k$ joining each pair of vertices in the graph. In particular, if $B$ is the adjacency matrix of a cycle of order $n$ we call the corresponding graphs graphs with cyclic defect or excess; these graphs are the subject of our attention in this paper. We prove the non-existence of infinitely many such graphs. As the highlight of the paper we provide the asymptotic upper bound of $O(\frac{64}3d^{3/2})$ for the number of graphs of odd degree $d\ge3$ and cyclic defect or excess. This bound is in fact quite generous, and as a way of illustration, we show the non-existence of some families of graphs of odd degree $d\ge3$ and cyclic defect or excess. Actually, we conjecture that, apart from the Möbius ladder on 8 vertices, no non-trivial graph of any degree $\ge 3$ and cyclic defect or excess exists.

scholarly journals Analysis of Inclined Strip Anchors in Sand Based on the Block Set Mechanism

2014 ◽  
Vol 553 ◽  
pp. 422-427 ◽  
Author(s):  
S.B. Yu ◽  
J.P. Hambleton ◽  
Scott William Sloan
Keyword(s):  
Upper Bound ◽  
Limit Analysis ◽  
Experimental Studies ◽  
Collapse Mechanism ◽  
Uplift Capacity ◽  
Collapse Mechanisms ◽  
Kinematic Method ◽  
The Subject ◽  
Uplift Resistance ◽  
Modelling Assumptions

Anchors are widely used in foundation systems for structures requiring uplift resistance. As demonstrated by numerous theoretical and experimental studies on the subject, uncertainty remains as to both the theoretical uplift capacity of anchors in idealised soils and the suitability of the various modelling assumptions in capturing the responses observed during tests. This study, which deals exclusively with the theoretical uplift capacity, presents newly obtained predictions of uplift capacities and the corresponding collapse mechanisms for inclined strip anchors in sand. The analysis is based on the upper bound (kinematic) method of limit analysis and the so-called block set mechanism, in which a collapse mechanism consisting of sliding rigid blocks is optimised with respect to interior angles and edges of the blocks. It is demonstrated that the method provides lower (better) estimates of uplift capacity in some cases compared to previous upper bound calculations. Also, variations in the predicted collapse mechanism with changes in embedment and inclination are assessed in detail.

scholarly journals On bounds for steady waves with negative vorticity

2021 ◽  
Vol 23 (2) ◽  
Author(s):  
Evgeniy Lokharu
Keyword(s):  
Froude Number ◽  
Solitary Waves ◽  
Upper Bound ◽  
Critical Value ◽  
Two Dimensional ◽  
Absolute Value ◽  
Constant Vorticity ◽  
Negative Constant

AbstractWe prove that no two-dimensional Stokes and solitary waves exist when the vorticity function is negative and the Bernoulli constant is greater than a certain critical value given explicitly. In particular, we obtain an upper bound $$F \le \sqrt{2} + \epsilon $$ F ≤ 2 + ϵ for the Froude number of solitary waves with a negative constant vorticity, sufficiently large in absolute value.

scholarly journals The Maximal Prime Gaps Supremum and the Firoozbakht's Hypothesis No 30

2020 ◽  
Author(s):  
Jan Feliksiak
Keyword(s):  
Upper Bound ◽  
Prime Numbers ◽  
Heuristic Argument ◽  
Mathematical Problems ◽  
The Subject ◽  
Prime Gaps ◽  
Theoretical Results

The maximal prime gaps upper bound problem is one of the major mathematical problems to date. The objective of the current research is to develop a standard which will aid in the understanding of the distribution of prime numbers. This paper presents theoretical results which originated with a researchin the subject of the maximal prime gaps. the document presents the sharpest upper bound for the maximal prime gaps ever developed. The result becomes the Supremum bound on the maximal prime gaps and subsequently culminates with the conclusive proof of the Firoozbakht's Hypothesis No 30. Firoozbakht's Hypothesis implies quite a bold conjecture concerning the maximal prime gaps. In fact it imposes one of the strongest maximal prime gaps bounds ever conjectured. Its truth implies the truth of a greater number of known prime gaps conjectures, simultaneously, the Firoozbakht's Hypothesis disproves a known heuristic argument of Granville and Maier. This paper is dedicated to a fellow mathematician, the late Farideh Firoozbakht.

scholarly journals Learning the Concept of Absolute Value with Hawgent Dynamic Mathematics Software

2020 ◽  
Vol 16 (2) ◽  
pp. 160-169
Author(s):  
Jerito Pereira ◽  
Yongxing Huang ◽  
Jihe Chen ◽  
Neni Hermita ◽  
Maximus Tamur
Keyword(s):  
High School ◽  
Senior High School ◽  
Development Model ◽  
Absolute Value ◽  
Learning Mathematics ◽  
The Subject ◽  
Basic Concepts ◽  
The Absolute ◽  
Dynamic Mathematics ◽  
Mathematics Software

Understanding the concept is one aspect that is needed and must be owned by students in learning mathematics. This research aims to make learning media assisted by hawgent dynamic mathematics software on understanding the absolute value in grade 10 Senior high school to help teachers explain the concepts and assist students in finding and understanding the basic concepts of absolute value topic. The development model used in this research is ADDIE development model. ADDIE is an abridgment of Analyze, Design, Develop and Evaluation. Researchers the learning media Hawgent can help students to understand and find the concept of absolute value. Based on several aspects, clearly and attractively with an average of 78.9% in the excellent category. It is concluded that the development of learning media Hawgent dynamic mathematics software can be used on the subject of understanding the concept of absolute value and with the help of this software it can help teachers explain the concept of absolute value and the students are also very interested in this ICT-based learning media. This conclusion is related to the validation results of media experts and material experts, where the validation results from media experts on the use of learning media are in good categories.

scholarly journals Mean-square upper bound of Hecke $L$-functions on the critical line.

2003 ◽  
Vol Volume 26 ◽  
Author(s):  
A Sankaranarayanan
Keyword(s):  
Cusp Form ◽  
Upper Bound ◽  
Critical Line ◽  
Congruence Subgroup ◽  
Mean Square ◽  
Absolute Value ◽  
International Audience ◽  
The Mean ◽  
The Absolute

International audience We prove the upper bound for the mean-square of the absolute value of the Hecke $L$-functions (attached to a holomorphic cusp form) defined for the congruence subgroup $\Gamma_0 (N)$ on the critical line uniformly with respect to its conductor $N$.

scholarly journals Shortcomings of NPV Calculation: Does One Error Annul the Other?

2021 ◽  
Author(s):  
Nikolett Sereg
Keyword(s):  
Probability Distribution ◽  
Flow Pattern ◽  
Relative Error ◽  
Cash Flow ◽  
The Other ◽  
Accurate Approximation ◽  
Present Value ◽  
Absolute Value ◽  
Aggregate Effect ◽  
The Subject

The research aims at analyzing the aggregate effect of possible errors related to the "textbook" method of present value calculation. Two main errors could stem from end-of-period convention and calculation according to expected lifespan. The magnitude of such errors depends on the cash flow pattern and the probability distribution of asset life, therefore the combination that may be regarded as the most typical in practice has been chosen as the subject of the examination, i.e., the continuous exponential cash flow pattern with exponentially distributed life. Based on the result of previous studies examining the errors separately, it seems possible that the two errors lead to a more accurate approximation – considering the absolute value of the relative error – compared to making only one of the errors. After the examination, I conclude that in the most typical cases of practice, it is not worth to take either the correct cash flow pattern or the life uncertainty into account beyond the "textbook" method.

Sign in / Sign up

Export Citation Format

Share Document