This text addresses the use of Mathematica as a symbolic manipulator, a programming language and a general tool for knowledge representation. Also included is coverage of functional programming, rule-based programming, …
This text addresses the use of Mathematica as a symbolic manipulator, a programming language and a general tool for knowledge representation. Also included is coverage of functional programming, rule-based programming, procedural programming, object-oriented programming and graphics programming.
The publication in 1687 of Isaac Newton's monumental treatise Principia Mathematica has long been regarded as the event that ushered in the modern period in mathematical physics.Newton developed a set …
The publication in 1687 of Isaac Newton's monumental treatise Principia Mathematica has long been regarded as the event that ushered in the modern period in mathematical physics.Newton developed a set of techniques and methods based on a geometric form of the differential and integral calculus for dealing with point-mass dynamics; he further showed how the results obtained could be applied to the motion of the solar system.Other topics studied in the
Given a triple <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper S comma eta comma mu right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>S</mml:mi> <mml:mo>,</mml:mo> <mml:mi>η<!-- η --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>μ<!-- μ --></mml:mi> <mml:mo stretchy="false">)</mml:mo> …
Given a triple <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis upper S comma eta comma mu right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>S</mml:mi> <mml:mo>,</mml:mo> <mml:mi>η<!-- η --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>μ<!-- μ --></mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(S,\eta ,\mu )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on a category <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with equalizers, one can form a new triple whose functor <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q"> <mml:semantics> <mml:mi>Q</mml:mi> <mml:annotation encoding="application/x-tex">Q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the equalizer of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="eta upper S"> <mml:semantics> <mml:mrow> <mml:mi>η<!-- η --></mml:mi> <mml:mi>S</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\eta S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S eta"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>η<!-- η --></mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">S\eta</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Fakir has studied conditions for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Q"> <mml:semantics> <mml:mi>Q</mml:mi> <mml:annotation encoding="application/x-tex">Q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to be idempotent, that is, to determine a reflective subcategory of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Here we regard <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as the composition of an adjoint pair of functors and give several new such conditions. As one application we construct a reflector in an elementary topos <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> from an injective object <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I"> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding="application/x-tex">I</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, taking <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S equals upper I Superscript upper I Super Superscript left-parenthesis minus right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>I</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>I</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> </mml:mrow> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">S = {I^{{I^{( - )}}}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We show that this reflector preserves finite limits and that the sheaf reflector for a topology in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {A}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be obtained in this way. We also show that sheaf reflectors in functor categories can be obtained from a triple of the form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S equals upper I Superscript left-parenthesis minus comma upper I right-parenthesis Baseline comma upper I"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>I</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mo>,</mml:mo> <mml:mi>I</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi>I</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">S = {I^{( - ,I)}},I</mml:annotation> </mml:semantics> </mml:math> </inline-formula> injective, which we studied in a previous paper. We deduce that the opposite of a sheaf subcategory of a functor category is tripleable over Sets.
In (13), Quillen defines a higher algebraic K -theory by taking homotopy groups of the classifying spaces of certain categories. Certain questions in K -theory then become questions such as …
In (13), Quillen defines a higher algebraic K -theory by taking homotopy groups of the classifying spaces of certain categories. Certain questions in K -theory then become questions such as when do functors induce a homotopy equivalence of classifying spaces, or when is a square of categories homotopy cartesian? Quillen has given some techniques for answering such questions. F. Waldhausen has extended these ideas in (19), and broadened the range of applications to include geometric topology (20).
The purpose of this paper is to extend the continuation theorem of holomorphic functions, especially the generalized Hartogs-Osgood’s theorem given in the previous papers [6] and [8], to the case …
The purpose of this paper is to extend the continuation theorem of holomorphic functions, especially the generalized Hartogs-Osgood’s theorem given in the previous papers [6] and [8], to the case of sections of torsion-free coherent analytic sheaves over a reduced complex space. In the following, we restrict ourselves only to reduced complex spaces.
In the frame of the recent axiomatic theories of harmonic functions [2], [3], [1], it has been shown that the continuous bounded functions on the boundaries of relatively compact open …
In the frame of the recent axiomatic theories of harmonic functions [2], [3], [1], it has been shown that the continuous bounded functions on the boundaries of relatively compact open sets are resolutive [5], [1]. The aim of the present paper is to substitute in these results the continuous functions by Borel-measurable functions and to leave out the restriction that the open sets are relatively compact. H. Bauer has replaced the axiom 3 of Brelot’s axiomatic by two weaker axioms: the axiom of separation (Trennungsaxiom) and the axiom K 1 . Since the axiom of separation is not fulfilled in some important cases (e.g. the compact Riemann surfaces) we shall weaken this axiom too, substituting it by one of its consequences: the minimum principle for hyperharmonic functions.
Throughout this paper, (R, m) denotes a (noetherian) local ring R with maximal ideal m . In [5], Monsky and Washnitzer define weakly complete R -algebras with respect to m. …
Throughout this paper, (R, m) denotes a (noetherian) local ring R with maximal ideal m . In [5], Monsky and Washnitzer define weakly complete R -algebras with respect to m. In brief, an R -algebra A † is weakly complete if
INTRODUCTION. The theory of sheaves, as it is exposed in the classical book of R. Godement {2}, has been generalized in sucessive stops. Ending this process, M. Artin introduced the …
INTRODUCTION. The theory of sheaves, as it is exposed in the classical book of R. Godement {2}, has been generalized in sucessive stops. Ending this process, M. Artin introduced the notion of Grothendieck topology and developed the fundamental part of the theory in a functorial way (cf. {1}). Although, the concept of Grothen dieck topology seems to be insufficient to relate certain aspects of the theory of sheaves; for example, the notion .of subspace (not necessarily open !) is omited and so, induced sheaves and relative cohomology must be ignored.
Received by the editors 2004-10-30. Transmitted by Steve Lack, Ross Street and RJ Wood. Reprint published on 2005-04-23. Several typographical errors corrected 2012-05-13. 2000 Mathematics Subject Classification: 18-02, 18D10, 18D20.
Received by the editors 2004-10-30. Transmitted by Steve Lack, Ross Street and RJ Wood. Reprint published on 2005-04-23. Several typographical errors corrected 2012-05-13. 2000 Mathematics Subject Classification: 18-02, 18D10, 18D20.