It is common to characterize languages as being composed of words and rules (Pinker,
2000
), or lexicon and syntax. It turns out that the demarcation in grammar between lexicon and syntax is much more challenging than is ordinarily thought. We stated at the outset that a theory of argument realization must isolate the aspects of the meanings of verbs which determine (or correlate with) argument realization, it must determine the nature of lexical representation, and it must determine the nature of mapping between lexical representation and syntactic representation. As we have seen, there appears to be a broad consensus across all theories as to the elements of meaning which figure into argument realization, but the conception of what is to be taken as strictly lexical has undergone radical changes over the years. It should be stressed that a full account of argument realization will not apportion all properties of argument realization phenomena to the syntax and the lexicon. Argument realization properties in specific cases can be attributed to a wide range of factors which are just beginning to be teased apart. These include, in addition to properties attributed to lexical head and functional heads and their compositional properties, factors such as information structure and discourse principles, as demonstrated, for the causative alternation in Rappaport Hovav (
2014
) and the dative alternation in Arnold et al. (
2000
), Bresnan et al (
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) and Rappaport Hovav and Levin (
2008
).
Finally, it should be pointed out that there is a rich literature on cross-linguistic variation in argument realization, a topic not addressed here because of space limitations. Much of this literature focuses on what has come to be called ‘,’—patterns of how conceptual components of event descriptions are distributed across morpho-syntactic elements—beginning with Talmy (
1985
); see also the work of Slobin (
1996
,
1997
); and analyses in a generative framework of these data in Acedo-Matellán and Mateu (
2013
) This work has uncovered interesting cross-linguistic patterns and has developed a typology of languages that display different lexicalization patterns. In addition, there is a rich literature on cross-linguistic patterns of valency, as documented, for example in Mal’chukov and Comrie (
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).
In all areas of research having to do with the relation of words and their morpho-syntactic environments, the solutions to certain problems bring into focus new ones and the area of argument realization remains a fertile for research.
5. Summary and Prospectus
The finite element method has many concepts and a jungle of details. This learning strategy minimizes the mixing of ideas, concepts, and technical details.
General idea of finding an approximation u(x) to some given f(x) :
u(x) = \sum_{i=0}^N c_i\baspsi_i(x)
where
We shall address three approaches:
Our mathematical framework for doing this is phrased in a way such that it becomes easy to understand and use the
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software package for finite element computing.
Given a vector \f = (3,5) , find an approximation to \f directed along a given line.
V = \mbox{span}\,\{ \psib_0\}
Define
\begin{equation*} \frac{\partial E}{\partial c_0} = 0 \end{equation*}
\begin{align*} E(c_0) = (\e,\e) = (\f - \u, \f - \u) = (\f - c_0\psib_0, \f - c_0\psib_0)\\ = (\f,\f) - 2c_0(\f,\psib_0) + c_0^2(\psib_0,\psib_0) \end{align*}
\begin{equation} \frac{\partial E}{\partial c_0} = -2(\f,\psib_0) + 2c_0 (\psib_0,\psib_0) = 0 \tag{1} \end{equation}
c_0 = \frac{(\f,\psib_0)}{(\psib_0,\psib_0)} = \frac{3a + 5b}{a^2 + b^2}
Observation to be used later: the vanishing derivative
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can be alternatively written as
(\e, \psib_0) = 0
Given a vector \f , find an approximation \u\in V :
\begin{equation*} V = \hbox{span}\,\{\psib_0,\ldots,\psib_N\} \end{equation*}
We have a set of linearly independent basis vectors \psib_0,\ldots,\psib_N . Any \u\in V can then be written as
\u = \sum_{j=0}^Nc_j\psib_j
Idea: find c_0,\ldots,c_N such that E= ||\e||^2 is minimized, \e=\f-\u .
\begin{align*} E(c_0,\ldots,c_N) = (\e,\e) = (\f -\sum_jc_j\psib_j,\f -\sum_jc_j\psib_j) \nonumber\\ = (\f,\f) - 2\sum_{j=0}^Nc_j(\f,\psib_j) + \sum_{p=0}^N\sum_{q=0}^N c_pc_q(\psib_p,\psib_q) \end{align*}
\begin{equation*} \frac{\partial E}{\partial c_i} = 0,\quad i=0,\ldots,N \end{equation*}
After some work we end up with a
\begin{align} \sum_{j=0}^N A_{i,j}c_j = b_i,\quad i=0,\ldots,N \tag{2}\\ A_{i,j} = (\psib_i,\psib_j) \tag{3}\\ b_i = (\psib_i, \f) \tag{4} \end{align}
Can be shown that minimizing ||\e|| implies that \e is orthogonal to all \v\in V :
(\e,\v)=0,\quad \forall\v\in V
which implies that \e most be orthogonal to each basis vector:
(\e,\psib_i)=0,\quad i=0,\ldots,N
This orthogonality condition is the principle of the projection (or Galerkin) method. Leads to the same linear system as in the least squares method.
Let V be a spanned by a set of \baspsi_0,\ldots,\baspsi_N ,
\begin{equation*} V = \hbox{span}\,\{\baspsi_0,\ldots,\baspsi_N\} \end{equation*}
Find u\in V as a linear combination of the basis functions:
u = \sum_{j\in\If} c_j\baspsi_j,\quad\If = \{0,1,\ldots,N\}
As in the vector case, minimize the (square) norm of the error, E , with respect to the coefficients c_j , j\in\If :
E = (e,e) = (f-u,f-u) = \left(f(x)-\sum_{j\in\If} c_j\baspsi_j(x), f(x)-\sum_{j\in\If} c_j\baspsi_j(x)\right)
\frac{\partial E}{\partial c_i} = 0,\quad i=\in\If
But what is the scalar product when \baspsi_i is a function?
(f,g) = \int_\Omega f(x)g(x)\, dx
(natural extension from Eucledian product (\u, \v) = \sum_j u_jv_j )
\begin{align*} E(c_0,\ldots,c_N) = (e,e) = (f-u,f-u) \\ = (f,f) -2\sum_{j\in\If} c_j(f,\baspsi_i) + \sum_{p\in\If}\sum_{q\in\If} c_pc_q(\baspsi_p,\baspsi_q) \end{align*}
\frac{\partial E}{\partial c_i} = 0,\quad i=\in\If
The computations are , and consequently we get a linear system
\sum_{j\in\If} A_{i,j}c_j = b_i,\ i\in\If,\quad A_{i,j} = (\baspsi_i,\baspsi_j),\ b_i = (f,\baspsi_i)
As before, minimizing (e,e) is equivalent to
(e,\baspsi_i)=0,\quad i\in\If \tag{5}
which is equivalent to
(e,v)=0,\quad\forall v\in V \tag{6}
which is the projection (or Galerkin) method.
The algebra is the same as in the multi-dimensional vector case, and we get the same linear system as arose from the least squares method.
Approximate a parabola f(x) = 10(x-1)^2 - 1 by a straight line.
\begin{equation*} V = \hbox{span}\,\{1, x\} \end{equation*}
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If two straight lines cut one another, then they will make the angles at the point of section equal to four right angles.
It has been suggested that the definitions were added to the Elements sometime after Euclid wrote them. Another possibility is that they are actually from a different work, perhaps older. In Def.I.22 special kinds of quadrilaterals are defined including square, oblong (a rectangle that are not squares), rhombus (equilateral but not a square), and rhomboid (parallelogram but not a rhombus). Except for squares, these other shapes are not mentioned in the Elements. Euclid does use parallelograms, but they’re not defined in this definition. Also, the exclusive nature of some of these terms—the part that indicates not a square—is contrary to Euclid’s practice of accepting squares and rectangles as kinds of parallelograms.
In proposition
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(but nowhere else) angles with curved sides are compared with rectilinear angles which shows that rectilinear angles are to be considered as a special kind of plane angle. That agrees with Euclid’s definition of them in
I.Def.9
and
I.Def.8
.
Also in Book III, parts of circumferences of circles, that is, arcs, appear as magnitudes. Only arcs of equal circles can be compared or added, so arcs of equal circles comprise a kind of magnitude, while arcs of unequal circles are magnitudes of different kinds. These kinds are all different from straight lines. Whereas areas of figures are comparable, different kinds of curves are not.
Book V includes the general theory of ratios. No particular kind of magnitude is specified in that book. It may come as a surprise that ratios do not themselves form a kind of magnitude since they can be compared, but they cannot be added. See the guide on Book V for more information.
Number theory is treated in Books VII through IX. It could be considered that numbers form a kind of magnitude as pointed out by Aristotle.
Beginning in Book XI, solids are considered, and they form the last kind of magnitude discussed in the Elements.
Some of the propositions are constructions. A construction depends, ultimately, on the constructive postulates about drawing lines and circles. The first part of a proof for a constructive proposition is how to perform the construction. The rest of the proof (usually the longer part), shows that the proposed construction actually satisfies the goal of the proposition. In the list of propositions in each book, the constructions are displayed in red.
Most of the propositions, however, are not constructions. Their statements say that under certain conditions, certain other conditions logically follow. For example, Prop.I.5 says that if a triangle has the property that two of its sides are equal, then it follows that the angles opposite these sides (called the “base angles”) are also equal. Even the propositions that are not constructions may have constructions included in their proofs since auxiliary lines or circles may be needed in the explanation. But the bulk of the proof is, as for the constructive propositions, a sequence of statements that are logically justified and which culminates in the statement of the proposition.
items $0.00
by: Debbie D. Philip (Author)
add to cart Login to save this to your wish listThousands of Chinese students visit our churches and join Christian activities. Many even say they have become Christians while abroad. Some go on to make great contributions to Chinese church and society. Sadly, however, many fall away after they return to China. Debbie Philip has visited hundreds of returnees. She offers a new perspective for understanding what happens when Chinese students encounter Christians abroad and what needs to happen if they are to continue following Christ after returning home. The life stories, illustrations, and suggestions in this book will help you understand and support Chinese returnees better as they prepare to go home.
Debbie Philip blends the hues of lengthy personal experience in ministry among Chinese students, in-depth research, more than 100 visits to returnees in China, relevant Biblical passages, wise and gentle recommendations, and articulate descriptions of a multi-faceted framework in presenting rich and unique portraits of the realities and challenges, as well as the necessary preparations that Chinese believers studying abroad should consider. The portraits include both a composite “generic” Chinese student reflecting the Chinese contexts, and seven real Chinese student stories. This gift is equally valuable for host-country friends of the Chinese academic community who wish to help with reentry preparation. Non-Chinese readers will appreciate a readable and succinct overview about the various Chinese contexts and culture. This treasure is for all—Chinese and everyone else.
former Lausanne Catalyst for International Student Ministries and former president of the Association of Christians Ministering among Internationals (ACMI)
Dr. Debbie Philip’s book is an invaluable addition to the body of literature on Chinese returnees. The stories from her research shed light on the important issues and concerns that International Student Ministry (ISM) workers need to know and address in order to provide the best assistance to returning Chinese graduates and scholars. By painting the historical and cultural contexts that the students are coming from and returning to, readers have a fuller appreciation of the specific chal¬lenges Chinese returnees face. I am impressed with Dr. Philip's humble recognition of the need for global partnership among churches, ministries and organizations in providing a more complete reentry service. This book is a testament to the power of the Gospel to change international student lives. It is a must-read book.