... EFFORT AND EXPENSE HAVE GONE INTO THE
DEVELOPMENT AND DOCUMENTATION OF SAP2000. THE PROGRAM HAS
BEEN THOROUGHLY TESTED AND USED. IN USING THE PROGRAM,
HOWEVER, THE USER ACCEPTS AND UNDERSTANDS ...
1 - 2 Overview of the Program
1
Overview of the Program
SAP2000 is a stand-alone finite-element-based structural program for the
analysis and design of civil structures. It offers an intuitive, ... spans
60 feet, and has a width and height of 12 feet each. The supports are roll-
ers at one end, and pins at the other.
The trusses and cross members are to be constructed of 2L4X4’s, while...
... Dimensions and tolerances – merge, selection, and snap
tolerances; font sizes; zoom increment; and others
Design codes and their parameters
Colors of objects and results for display and printing ... i.e., as data organized in a set of
tables with specified table names and column headings. These data
include the definition of the model and the results of analysis and design.
Tabular data can ...
objects and elements.
Groups
A group is a named collection of objects. It may contain any number of
objects of any number of types. Groups have many uses, including:
Quick selection of objects...
... Recently, there are a large number of papers dealing with the existence of
solutions ofnonlinear fractional differentialequations by the use of techniques of
nonlinear analysis (fixed point theore ... system of nonlinear
fractional differential equations. Int J Differ Equ 2010, 2010:1-12.
28. Ahmad B, Nieto JJ: Existence results for a coupled system ofnonlinear fractional differentialequations ... existence and uniqueness of solutions for initial value problem of nonlinear
fractional differential equations. 2010 Sixth IEEE/ASME International Conference on Mechatronic and Embedded Systems
and...
... biotechnology, and economics. For an introduction of the basic theory of impulsive
differential equations, see Lakshmikantham et al. 1; for an overview of existing results and
of recent research areas of ... problems,”
Journal of Mathematical Analysis and Applications, vol. 116, no. 2, pp. 415–426, 1986.
41 Z. Bai and H. Wang, “On positive solutions of some nonlinear fourth-order beam equations, ” Journal
of Mathematical ... hinged ends and many
nonlocal boundary conditions and established excellent existence results for multiple positive
solutions of fourth-order nonlinearequations which model deflections of an elastic...
... Wang, and L. Ge, “Existence of nonoscillatory solution of high-order
nonlinear difference equation,” Nonlinear Dynamics and Systems Theory, vol. 6, no. 2, pp. 205–210, 2006.
33 E. M. E. Zayed and ... behaviour of solutions of second- and third-order nonlinear
differential equations in recent years. See, for instance, 1–37 as some related papers or
Journal of Inequalities and Applications 7
Proof. ... J C. Liang, and D H. Wang, “Existence of nonoscillatory solution of
third order linear neutral delay difference equation with positive and negative coefficients,” Nonlinear
Dynamics and Systems...
... analysis (FA) and partial differentialequations (PDEs). The first
part deals with abstract results in FA and operator theory. The second part concerns
the study of spaces of functions (of one or ... They appear in linearandnonlinear PDEs that arise, for
example, in differential geometry, harmonic analysis, engineering, mechanics, and
physics. They belong to the toolbox of any graduate ... {x
0
} and C.
Proof of Lemma 1.3. After a translation we may always assume that 0 ∈ C.We
may thus introduce the gauge p of C (see Lemma 1.2). Consider the linear subspace
G = Rx
0
and the linear...
... the general solution of the differential equation.
Chapter 0 Ordinary DifferentialEquations 3
Principle of Superposition.
If u
1
(t) and u
2
(t) are solutions of the same linear
homogeneous equation ... acknowledge the skillful work of Cindy Smith, who was the LaTeX
compositor and corrected many of my mistakes, the help of Academic Press
editors and consultants, and the guidance of reviewers for this ... follow the derivations of the heat and wave
equations.
The principal objective of the book is solving boundary value problems
involving partial differential equations. Separation of variables receives...
... connected subset Σ of R
n
(Σ is the state space and elements of Σ are
called states)
and
(b) two C
2
functions
T :Σ→ (0, ∞)
Λ:Σ→ R
n
.
T is the temperature and the components of Λ are the generalized ... equilibrium
B. Examples
1. Simple fluids
2. Other examples
C. Physical interpretations of the model
1. Equilibrium
2. Positivity of temperature
3. Extensive and intensive parameters
4. Concavity of S
5. ... Entropy and irreversibility
A. A model material
1. Definitions
2. Energy and entropy
a. Working and heating
b. First Law, existence of E
c. Carnot cycles
d. Second Law
e. Existence of S
3. Efficiency of...
... an offshoot of the investigation of the corresponding
questions for nonlinear waves. The nonlinear problems required a more
powerful approach altogether, and eventually the possibility of using ... in pressure, density, and velocity: the
blast waves of explosions and the sonic booms of high speed aircraft. But
the whole intricate machinery ofnonlinear hyperbolic equations had to be
developed ... in gas dynamics led
to most of the fundamental ideas in nonlinear hyperbolic waves. The most
outstanding new phenomenon of the nonlinear theory is the appearance of
shock waves, which are abrupt...
... Lower and Upper Solutions
Another modification of the concepts of lower and upper solutions concerns the possibility of
allowing jumps in their graphs. Since the task of finding a pair of lower and ... of α and β and r ≤ x ≤ 1/r.
Therefore Theorem 5.4 implies the existence of a weak solution of 7.7 between α and
β. Moreover, since f is positive between α and β the solution is increasing and, ... ε, t
0
L, and since
ε ∈ 0,L was fixed arbitrarily, the proof of Step 4 is complete.
Conclusion
The construction of y
∞
and Step 1 imply that y
∞
≥ x
∗
and the definition of x
∗
and Step 4
imply...
... 5.6 gives
a characterization of a class ofnonlinear periodic problems 5.1a and 5.1b which have
only one solution. We begin the investigation of problem 5.1a and 5.1b with a uniqueness
result.
Theorem ... discontinuous for certain values of
the time variable t ∈ 0,T.Thisformoff is motivated by a variety of initial and boundary
value problems known from applications and having nonlinear, discontinuous forcing ... in both, the case of intersecting solutions u
1
and
u
2
and the case of separated u
1
and u
2
, there exists an interval α, β ⊂ 0,T satisfying 5.5.
Now, by 5.1a, 5.3 ,and 5.5,weobtain
v
t
>
a
t
−...
... conditions of Theorem 1.6.
4. Examples
From the main results of the present paper new and optimal in some sense sufficient condi-
tions for the existence of periodic solutions oflinearand sublinear ... one and only one ω-periodic solution.
10 Boundary Value Problems
3. Proof of the main results
Proof of Theorem 1.1. Without loss of generality, it can be assumed that h00. On the other
hand, ... Gaines and J. L. Mawhin, Coincidence Degree, andNonlinear Differential Equations, vol.56ofLecture
Notes in Mathematics, Springer, Berlin, Germany, 1977.
5 L. A. Kipnis, “On periodic solution of...
... prototypes for the development of the
basic theory of the global behavior ofnonlinear difference equationsof order greater than
one come from the results of rational difference equations. For the systematical ... Open
Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2002.
2 R. P. Agarwal, Difference Equationsand Inequalities. Theory, Methods, and Applications, vol. 155 of Mono-
graphs and ... and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1992.
3 V. L. Koci
´
c a nd G. Ladas, Global Behavior ofNonlinear D i fference Equationsof Higher Order with...