scholarly journals There are no n-point Fσ sets in Rm

2005 ◽  
Vol 72 (3) ◽  
pp. 477-480 ◽  
Author(s):  
David L. Fearnley ◽  
L. Fearnley ◽  
J. W. Lamoreaux
Keyword(s):  
Natural Extension ◽  
Affine Hyperplane ◽  
Positive Integers

We show that, for any positive integers n and m, if a set S ⊂ Rm intersects every m − 1 dimensional affine hyperplane in Rm in exactly n points, then S is not an Fσ set. This gives a natural extension to results of Khalid Bouhjar, Jan J. Dijkstra, and R. Daniel Mauldin, who have proven this result for the case when m = 2, and also Jan J. Dijkstra and Jan van Mill, who have shown this result for the case when n = m.

THE DOT-DEPTH OF A GENERATING CLASS OF APERIODIC MONOIDS IS COMPUTABLE

1992 ◽  
Vol 03 (04) ◽  
pp. 419-442 ◽  
Author(s):  
F. BLANCHET-SADRI
Keyword(s):  
Natural Extension ◽  
Finite Alphabet ◽  
Aperiodic Monoids ◽  
Positive Integers

Given a finite alphabet A and a sequence of positive integers [Formula: see text] congruences on A*, denoted by [Formula: see text] and related to a version of the Ehrenfeucht-Fraïssé game, have been defined by Thomas in order to give a new proof that the Brzozowski’s dot-depth hierarchy of star-free languages is infinite. A natural extension of some of the results of Thomas states that the monoid variety corresponding to level k of the Straubing hierarchy (the Straubing hierarchy is closely related to the Brzozowski’s dot-depth hierarchy) can be characterized in terms of the monoids [Formula: see text]. In this paper, it is shown that the dot-depth of the [Formula: see text]’s is computable.

High-resolution electron microscopy of Cu2O surfaces

1986 ◽  
Vol 44 ◽  
pp. 396-397
Author(s):  
V. Castano ◽  
W. Krakow
Keyword(s):  
High Resolution ◽  
Natural Extension ◽  
High Resolution Electron Microscopy ◽  
Point Of View ◽  
Initial Oxidation ◽  
System A ◽  
Resolution Electron ◽  
Surface Reconstructions ◽  
Atomic Surface ◽  
Au Thin Films

In non-UHV microscope environments atomic surface structure has been observed for flat-on for various orientations of Au thin films and edge-on for columns of atoms in small particles. The problem of oxidation of surfaces has only recently been reported from the point of view of high resolution microscopy revealing surface reconstructions for the Ag2O system. A natural extension of these initial oxidation studies is to explore other materials areas which are technologically more significant such as that of Cu2O, which will now be described.

Universal ESEM

1993 ◽  
Vol 51 ◽  
pp. 786-787
Author(s):  
G.D. Danilatos
Keyword(s):  
Electron Microscope ◽  
Scanning Electron Microscope ◽  
High Vacuum ◽  
Natural Extension ◽  
Necessary Condition ◽  
Environmental Scanning ◽  
Detection Systems ◽  
Small Gain ◽  
Detection Medium ◽  
Scanning Electron

The environmental scanning electron microscope (ESEM) has evolved as the natural extension of the scanning electron microscope (SEM), both historically and technologically. ESEM allows the introduction of a gaseous environment in the specimen chamber, whereas SEM operates in vacuum. One of the detection systems in ESEM, namely, the gaseous detection device (GDD) is based on the presence of gas as a detection medium. This might be interpreted as a necessary condition for the ESEM to remain operational and, hence, one might have to change instruments for operation at low or high vacuum. Initially, we may maintain the presence of a conventional secondary electron (E-T) detector in a "stand-by" position to switch on when the vacuum becomes satisfactory for its operation. However, the "rough" or "low vacuum" range of pressure may still be considered as inaccessible by both the GDD and the E-T detector, because the former has presumably very small gain and the latter still breaks down.

The Standard Metric

2017 ◽  
Author(s):  
Bernhard M¨uhlherr ◽  
Holger P. Petersson ◽  
Richard M. Weiss
Keyword(s):  
Affine Transformation ◽  
Convex Sets ◽  
Finite Dimension ◽  
Affine Subspace ◽  
Affine Transformations ◽  
Real Vector ◽  
Euclidean Metric ◽  
Affine Hyperplane ◽  
Tits Building ◽  
Convex Closure

This chapter presents some results about groups generated by reflections and the standard metric on a Bruhat-Tits building. It begins with definitions relating to an affine subspace, an affine hyperplane, an affine span, an affine map, and an affine transformation. It then considers a notation stating that the convex closure of a subset a of X is the intersection of all convex sets containing a and another notation that denotes by AGL(X) the group of all affine transformations of X and by Trans(X) the set of all translations of X. It also describes Euclidean spaces and assumes that the real vector space X is of finite dimension n and that d is a Euclidean metric on X. Finally, it discusses Euclidean representations and the standard metric.

Language, procedures, and the non-perceptual origin of natural number concepts

2016 ◽  
Author(s):  
David Barner
Keyword(s):  
General Framework ◽  
Ad Hoc ◽  
Building Blocks ◽  
Number Words ◽  
Linguistic Markers ◽  
Logical Foundations ◽  
Large Numbers ◽  
Perceptual Representations ◽  
Positive Integers ◽  
Perceptual Systems

Perceptual representations – e.g., of objects or approximate magnitudes –are often invoked as building blocks that children combine with linguisticsymbols when they acquire the positive integers. Systems of numericalperception are either assumed to contain the logical foundations ofarithmetic innately, or to supply the basis for their induction. Here Ipropose an alternative to this general framework, and argue that theintegers are not learned from perceptual systems, but instead arise toexplain perception as part of language acquisition. Drawing oncross-linguistic data and developmental data, I show that small numbers(1-4) and large numbers (~5+) arise both historically and in individualchildren via entirely distinct mechanisms, constituting independentlearning problems, neither of which begins with perceptual building blocks.Specifically, I propose that children begin by learning small numbers(i.e., *one, two, three*) using the same logical resources that supportother linguistic markers of number (e.g., singular, plural). Several yearslater, children discover the logic of counting by inferring the logicalrelations between larger number words from their roles in blind countingprocedures, and only incidentally associate number words with perception ofapproximate magnitudes, in an *ad hoc* and highly malleable fashion.Counting provides a form of explanation for perception but is not causallyderived from perceptual systems.

أوهام القراءة الحداثية المعاصرة: دراسة تحليلية نقدية (Misunderstandings of Modern Contemporary Reading: A Critical Analysis )

2018 ◽  
Vol 15 (3) ◽  
pp. 258-299
Author(s):  
حنان ساري ◽  
محمد أبو الليث الخيرآبادي
Keyword(s):  
Critical Analysis ◽  
Natural Extension ◽  
Modern Times ◽  
The Past ◽  
Religious Issues ◽  
European Experience ◽  
Islamic Thought ◽  
Critical Approaches ◽  
Muslim Scholars

انتشرت لفظة الحداثة في عصرنا الحالي انتشاراً واسعاً، وأخذت مفهومات متعددة، ونحن لا نراها أكثر من أنها امتداد طبيعي للقلق الأوروبي.وسعى التيار الحداثي لتقديم مشاريع تعتمد كلية على مناهج وآليات غربية في دراستها وتعاملها مع القرآن الكريم والسنة، ولعل أهم الذين تقدموا بتلك المشاريع؛ محمد أركون، عبد المجيد الشرفي التونسي، محمد عابد الجابري، حسن حنفي، نصر حامد أبوزيد، الطيب التيزني السوري، محمد شحرور، جمال البنا وغيرهم، وطالبوا بإعادة قراءة القرآن الكريم على ضوء المناهج النقدية الغربية في عملية التقليد الأعمى، ومن ثم نقلوا التجربة الأوروبية بكل آثارها الفوضوية إلى ساحة الفكر الإسلامي. وإن مدعي تجديد الدين من هؤلاء، ليس لهم صلة بالدين أو علومه، بقدر ما تشبعت أفكارهم بمناهج علمانية، فالمراد من جهودهم ليس الدين، وإنما غرس الحداثة بدل الدين، فهي خطَّةٌ تقوم على التَّغيير من داخل البيت الإسلاميِّ من خلال العبث بالنُّصوص الشَّرعيَّة بتحريفها وتفريغها من محتواها الحقيقيِّ، ووضع المحتوى الذي يريدون؛ فهم يَطرحون أفكارَهم وآراءَهم على أنَّها رؤى إسلاميَّة ناشئة عن الاجتهاد في فهم الدِّين. وقد حَمَلَ هذا الاتجاهُ شعار (التَّحديث والعصرنة للإسلام)؛ فهم يريدون منَّا تركَ ما أَجْمَعَتْ عليه الأُمَّةُ من معاني القرآن والسُّنَّة، لفهم جديد مغاير لفهم السَّلَف الصَّالح يكون متناسبًا مع هذا العصر الذي نعيش فيه. الكلمات المفتاحيّة: الحداثة، أوهام، الحداثيون، قراءة معاصرة، العصرنة للإسلام. Abstract In modern times, the word Modernity has spread widely and has become widely understood, and we see it as a natural extension of European concern and confusion. The Modernist Movement strived to present the ideas that rely completely on Western methodologies and approaches in their study and dealing with Qur’an and Sunnah. The most important scholars that have presented these ideas are; Mohammad Arkoun, ‘Abd Al-Majid Sharafi al-Tunisi, Mohammed ‘Abed al-Jabri, Hassan Hanafi, Nasr Hamid Abu Zayd, Tayyeb Tizini, Muhammad Shahrour, Jamal Al-Banna, and others, they called for a re-reading and reinterpret the Qur’an in the light of Western critical approaches. Then, conveyed and brought the European experience and practice with all its chaotic effects to Islamic thought. The slogan of “Renewal of Religion” from these people has no relation to religion (Islam) or its sources, but instead saturated their ideas with secular methods. They tried to instill modernity rather than religion, and misinterpreted the Islamic sources by distorting it and evacuating it from the true context and setting it with their own understanding. They claim their ideas and opinions as the effort to understand religion and carried the slogan of “Modernization and Modernization of Islam”; they want us to leave the consensus of the Muslim scholars on religious issues (Ijmaa’ al-Ummah) especially relating to the meaning of the Qur’an and Sunnah and bring us to a new views and understanding on religious issues which are contradictory to the views of the past Muslim scholars (al-salaf al-soleh) to fulfill their opinions. Keywords: Modernity, Misunderstanding, Modernists, Contemporary Reading, Modernization of Islam.

Short Generating Functions for some Semigroup Algebras

10.37236/1729 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Graham Denham
Keyword(s):  
Rational Function ◽  
Generating Functions ◽  
Hilbert Series ◽  
Simple Expression ◽  
Arbitrary Field ◽  
Graded Algebra ◽  
Semigroup Algebras ◽  
Positive Integers

Let $a_1,\ldots,a_n$ be distinct, positive integers with $(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let $H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any $n\geq4$.

On a Two-Sided Turán Problem

10.37236/1735 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Dhruv Mubayi ◽  
Yi Zhao
Keyword(s):  
Minimum Size ◽  
Turan Problem ◽  
Turan Problems ◽  
Turán Problem ◽  
Positive Integers

Given positive integers $n,k,t$, with $2 \le k\le n$, and $t < 2^k$, let $m(n,k,t)$ be the minimum size of a family ${\cal F}$ of nonempty subsets of $[n]$ such that every $k$-set in $[n]$ contains at least $t$ sets from ${\cal F}$, and every $(k-1)$-set in $[n]$ contains at most $t-1$ sets from ${\cal F}$. Sloan et al. determined $m(n, 3, 2)$ and Füredi et al. studied $m(n, 4, t)$ for $t=2, 3$. We consider $m(n, 3, t)$ and $m(n, 4, t)$ for all the remaining values of $t$ and obtain their exact values except for $k=4$ and $t= 6, 7, 11, 12$. For example, we prove that $ m(n, 4, 5) = {n \choose 2}-17$ for $n\ge 160$. The values of $m(n, 4, t)$ for $t=7,11,12$ are determined in terms of well-known (and open) Turán problems for graphs and hypergraphs. We also obtain bounds of $m(n, 4, 6)$ that differ by absolute constants.

A Note on Differential Identities in Prime and Semiprime Rings

2020 ◽  
pp. 77-83
Author(s):  
Mohammad Shadab Khan ◽  
Mohd Arif Raza ◽  
Nadeemur Rehman
Keyword(s):  
Prime Ring ◽  
Semiprime Ring ◽  
Nonzero Ideal ◽  
Differential Identities ◽  
Standard Identity ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings ◽  
Positive Integers

Let R be a prime ring, I a nonzero ideal of R, d a derivation of R and m, n fixed positive integers. (i) If (d ( r ○ s)(r ○ s) + ( r ○ s) d ( r ○ s)n - d ( r ○ s))m for all r, s ϵ I, then R is commutative. (ii) If (d ( r ○ s)( r ○ s) + ( r ○ s) d ( r ○ s)n - d (r ○ s))m ϵ Z(R) for all r, s ϵ I, then R satisfies s4, the standard identity in four variables. Moreover, we also examine the case when R is a semiprime ring.

Gauge Field Theory in Natural Geometric Language

2020 ◽  
Author(s):  
Daniel Canarutto
Keyword(s):  
Particle Physics ◽  
Quantum Physics ◽  
Brst Symmetry ◽  
Natural Extension ◽  
Integrated Approach ◽  
Arbitrary Spin ◽  
Field Theories ◽  
Quantum Field Theories ◽  
Standard View ◽  
Spinor Bundle

This monograph addresses the need to clarify basic mathematical concepts at the crossroad between gravitation and quantum physics. Selected mathematical and theoretical topics are exposed within a not-too-short, integrated approach that exploits standard and non-standard notions in natural geometric language. The role of structure groups can be regarded as secondary even in the treatment of the gauge fields themselves. Two-spinors yield a partly original ‘minimal geometric data’ approach to Einstein-Cartan-Maxwell-Dirac fields. The gravitational field is jointly represented by a spinor connection and by a soldering form (a ‘tetrad’) valued in a vector bundle naturally constructed from the assumed 2-spinor bundle. We give a presentation of electroweak theory that dispenses with group-related notions, and we introduce a non-standard, natural extension of it. Also within the 2-spinor approach we present: a non-standard view of gauge freedom; a first-order Lagrangian theory of fields with arbitrary spin; an original treatment of Lie derivatives of spinors and spinor connections. Furthermore we introduce an original formulation of Lagrangian field theories based on covariant differentials, which works in the classical and quantum field theories alike and simplifies calculations. We offer a precise mathematical approach to quantum bundles and quantum fields, including ghosts, BRST symmetry and anti-fields, treating the geometry of quantum bundles and their jet prolongations in terms Frölicher's notion of smoothness. We propose an approach to quantum particle physics based on the notion of detector, and illustrate the basic scattering computations in that context.

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