text:
# A dependable term-expansion mechanism

Logtalk provides a dependable term-expansion mechanism that gives the user full and fine-grained control on if, when, and how expansions are applied. This is accomplished by introducing the concept of hook object and by allowing expansion rules to be defined and applied without relying on multifile predicates.

A hook object is simply an object implementing the expanding built-in protocol that declares term_expansion/2 and goal_expansion/2 predicates. This allows sets of expansion rules to be independently loaded, used, and combined. As the expansion predicates are not declared as multifile predicates, loading a hook object have no consequence by itself on the subsequent loading of source files. So… how do we expand a source file? The first option is to declare in the source file itself the hook object that should be used to expand it:

:- set_logtalk_flag(hook, my_hook_object).

Note that set_logtalk_flag/1 directives are local to the entity or source file containing them. The second option is to use the hook/1 compiler option of the compilation and loading built-in predicates:

| ?- logtalk_load(source, [hook(my_hook_object)]).
...

The third option is to define a default hook object using the set_logtalk_flag/1 predicate (which, unlike the directive, sets global and thus default flag values):

| ?- set_logtalk_flag(hook, my_hook_object).
...

By default, at startup, the hook flag is not defined. This means that, by default, no source file is expanded independently of any loaded hook objects.

Did you notice that only a single hook object can be set at any given time to expand a source file? What if we want to apply multiple expansions to a source file? In this case, you need to explicitly define how multiple hook objects are combined to define an expansion workflow. Different expansion workflow semantics are possible and required depending on the problem. For example, the expansions may be independent (that doesn’t necessarily mean, however, that the order the expansion rules are applied can be arbitrary or that their concurrent use is free of trouble…). The expansions may also be intended to be used as a well defined pipeline with the terms resulting from an expansion being passed to the next expansion. Or we may have a more complex workflow combining independent and linked expansions. To simplify defining expansion workflows, Logtalk provides hook_flows and hook_objects libraries with ready to use parametrizable workflows and hook objects.

Logtalk’s term-expansion mechanism may sound overly complex, specially for users of Prolog systems such as SWI-Prolog or YAP where the term_expansion/2 and goal_expansion/2 predicates are multifile and the common practice is just to dump the predicate definitions into user and wait for magic to happen. Any loaded library (by the user or the system itself) can contribute to a growing set of expansions, often without the user noticing. This work until conflicts arise between expansions. For example, consider the following module that counts the number of facts for the a/1 predicate and adds a fact with the count:

:- module(m1, []).

:- dynamic(counter_/1).

:- multifile(user:term_expansion/2).
:- dynamic(user:term_expansion/2).

user:term_expansion(begin_of_file, begin_of_file) :-
retractall(counter_(_)),
assertz(counter_(0)).
user:term_expansion(a(X), a(X)) :-
retract(counter_(Old)),
New is Old + 1,
assertz(counter_(New)).
user:term_expansion(end_of_file, [total_a(Total), end_of_file]) :-
counter_(Total).

Assume now a source.pl file with the following contents:

a(2). a(3). a(5).
b(7).

If we load the m1 module followed by the source file we get:

?- [m1].
true.

?- [file].
true.

?- total_a(Total).
Total = 3.

The expansion works as intended. But then we, or someone else, decide to define an expansion that counts b/1 predicate facts:

:- module(m2, []).

:- dynamic(counter_/1).

:- multifile(user:term_expansion/2).
:- dynamic(user:term_expansion/2).

user:term_expansion(begin_of_file, begin_of_file) :-
retractall(counter_(_)),
assertz(counter_(0)).
user:term_expansion(b(X), b(X)) :-
retract(counter_(Old)),
New is Old + 1,
assertz(counter_(New)).
user:term_expansion(end_of_file, [total_b(Total), end_of_file]) :-
counter_(Total).

Now we get with both expansions loaded:

?- [m1, m2].
true.

?- [source].
true.

?- total_a(Total).
Total = 3.

?- total_b(Total).
Correct to: "total_a(Total)"? no
ERROR: Unknown procedure: total_b/1
ERROR:   However, there are definitions for:
ERROR:         total_a/1
ERROR:
ERROR: In:
ERROR:   [10] total_b(_22126)
ERROR:    [9] <user>
Exception: (10) total_b(_11122) ? abort
% Execution Aborted

Ouch! The expansions code look sensible but loading the first module prevents the expansions defined in the second module from working as intended. Can you debug the problem and explain why? This is, of course, a toy example. In practice, expansions are often more complex and can be harder to debug when conflicts arise. It’s instructive to reimplement the above example using Logtalk:

:- object(o1,
implements(expanding)).

:- private(counter_/1).
:- dynamic(counter_/1).

term_expansion(begin_of_file, begin_of_file) :-
retractall(counter_(_)),
assertz(counter_(0)).
term_expansion(a(X), a(X)) :-
retract(counter_(Old)),
New is Old + 1,
assertz(counter_(New)).
term_expansion(end_of_file, [total_a(Total), end_of_file]) :-
counter_(Total).

:- end_object.

:- object(o2,
implements(expanding)).

:- private(counter_/1).
:- dynamic(counter_/1).

term_expansion(begin_of_file, begin_of_file) :-
retractall(counter_(_)),
assertz(counter_(0)).
term_expansion(b(X), b(X)) :-
retract(counter_(Old)),
New is Old + 1,
assertz(counter_(New)).
term_expansion(end_of_file, [total_b(Total), end_of_file]) :-
counter_(Total).

:- end_object.

We now get:

?- {hook_flows(loader)}.
...
true.

?- {o1, o2}.
...
true.

...
true.

?- total_a(Total).
Total = 3.

?- total_b(Total).
Total = 1.

The hook_pipeline/1 parametric object takes as parameter a list of hook objects and defines a pipeline of expansions where the results of an expansion by a hook object are passed for further expansion by the next hook object. Problem solved. Could the problem have been avoided in the first place? Note that expansions, specially when provided by libraries (system or third-party) are often written independently and by different people. Thus, solving conflicts often requires the definition of an explicit expansion workflow instead of relying on the default system workflow.

Does the conflicts only occur with term-expansion? What about goal-expansion? As an example, assume that we want to replace goals such as X is X0 + 1 with calls to the de facto standard succ/2 predicate:

:- module(succ_expansion, []).

:- multifile(user:goal_expansion/2).
:- dynamic(user:goal_expansion/2).

user:goal_expansion(X is X0 + 1, succ(X0,X)).

We may have a second expansion rule in another module that recognizes ground X is X0 + 1 goals and replaces them with either true or fail goals:

:- module(is_optimization, []).

:- multifile(user:goal_expansion/2).
:- dynamic(user:goal_expansion/2).

user:goal_expansion(X is X0 + 1, Goal) :-
ground(X is X0 + 1),
(X is X0 + 1 -> Goal = true; Goal = fail).

If the expansion rule in the succ_expansion module is applied first, the expansion rule in the is_optimization module will never be applied. The reverse order would work. But relying on the order expansions are applied only works reliable when we have full control of the order by using an explicit expansion workflow.

Can we emulate in Logtalk the convenience of the term-expansion mechanism as found in e.g. SWI-Prolog? It’s a fair question and the answer is yes. We can set the default hook object to user and define multifile term_expansion/2 and goal_expansion/2 predicates disregarding the expanding built-in protocol. It’s also possible to use the more structured Logtalk implementation of term-expansion while defining workflows with steps that use expansion rules defined in modules (see e.g. the prolog_module_hook/1 parametric hook object provided by the hook_objects library).

P.S. When comparing with other term-expansion implementations, we implicitly focused here only on Prolog systems whose term-expansion mechanism is derived from the original Quintus Prolog mechanism and based on the definition of term_expansion/2 and goal_expansion/2 multifile predicates. This mechanism is found on SWI-Prolog and YAP as mentioned. What about other Prolog systems? Similar to the original Quintus Prolog implementation, XSB only supports the term_expansion/2 predicate. SICStus Prolog uses predicates with the same names but with additional arguments to better handling passing layout information. GNU Prolog only provides limited support for the term_expansion/2 predicate. A few systems, notably Ciao, ECliPSe, and Qu-Prolog provide different support for expanding terms. As this short overview makes clear, there isn’t any de facto standard Prolog term-expansion mechanism. Moreover, only some Prolog systems provide a term-expansion mechanism.


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# How to route a USB MIDI Keyboard to a JACK-MIDI Input, under Debian.

One of the possessions which I have is a USB MIDI Keyboard, which I’d like to be able to play, so that my computer’s software synthesizers actually generate the sound…

I know that this can be done because I’ve done it before. But in the past, when I set this up, I was either using an ‘ALSA’ MIDI input, belonging to an ‘ALSA’ or ‘PulseAudio’ application such as “Linux Multimedia Studio”, or I was using ‘QSynth’, which is a graphical front-end to ‘fluidsynth’, but in such a way that QSynth was listening for ALSA MIDI, and outputting JACK audio. This is actually a very common occurrence. I can switch between using the ‘PulseAudio’ and using the ‘JACK’ sound daemon, through a carefully set-up configuration of ‘QJackCtl’, which suspends PulseAudio when I activate JACK, and which also resumes PulseAudio, when I shut down JACK again.

But there is a basic obstacle, as soon as I want to play my MIDI Keyboard through ‘Ardour’. Ardour v6 can be run with the PulseAudio sound system, but only for playback, or, Ardour can be run with its JACK sound back-end, after JACK has been launched. Ardour cannot be run with its ALSA back-end, when PulseAudio is running.

The default behaviour of the Debian kernel modules, when I plug in a USB MIDI Keyboard, is, to make that MIDI connection visible within my system as an ALSA MIDI source, even though some applications, such as Ardour, will insist on only taking input from JACK MIDI sources, when in fact running in JACK mode. And so, this problem needed to be solved this morning…

The solution which I found was, to feed the Keyboard, which happens to be an “Oxygen 61″, to the ‘MIDI Through Port’ that’s visible in the ALSA Tab of QJackCtl’s Connections window. When MIDI sequences are fed there, they are also output from the System JACK MIDI sources, visible in the MIDI Tab of QJackCtl’s Connections window:

I should also note that, in many cases, the JACK clients can ask the JACK sound daemon to be connected to various inputs and outputs from within, without absolutely requiring that the QJackCtl Connections window be used. This explains why the audio output of Ardour was already routed properly to my PC’s speakers. But I found that I could only keep track of the MIDI connection, through QJackCtl’s Connections window. As the screen-shots above show, the second step is, to feed one of the System Sources to the appropriate Ardour MIDI input, in the MIDI Tab of QjackCtl’s Connections window.

The result was, that the synthesizer which I have available as an Ardour plug-in, played beautifully, in response to my pressing keys on the actual MIDI Keyboard, and no longer just, when I clicked on the graphical keyboard within the Ardour application window:

This on-screen keyboard can be made visible, by double-Alt-Clicking on the icon of the instrument, with Ardour in its Mixer view, and then expanding the resulting windows’ MIDI Keyboard fly-out. Yet, the on-screen keyboard was only useful for setup and testing purposes.

(Updated 5/01/2021, 1h15… )

(As of 12/07/2020, 8h20: )

One fact which I should also mention is that, there exists the package ‘a2jmidid‘, which will solve the same sort of problem. When that package is installed – according to its package description – it causes a daemon to run, which will react to every ALSA, MIDI Input or Output port, by connecting to it, and creating a corresponding JACK MIDI Input or Output port.

That package was mainly meant to be used by people, who will entirely make MIDI connections from within their applications, and not using the ‘QJackCtl’ Connections window. And another big drawback of having that package installed will be, that it will automatically tie up all existing ALSA MIDI clients, to make them available to JACK… Therefore, that package can be counterproductive to install, if the user wants to switch back and forth, between having JACK running and not having it running.

(Update 12/07/2020, 17h20: )

I suppose that if a user is determined, to be switching back and forth between running ‘PulseAudio’ and ‘JACK’, and yet, wants the package ‘a2jmidid‘ to make all the ALSA-MIDI ports available as (forwarded) JACK-MIDI ports, then he or she can extrapolate on what I, myself have done, by adding more commands to ‘QJackCtl’. In the ‘Options’ tab, one can check the box named ‘Execute script after Startup’, and then in the field next to that tab, type:

a2jmidid -e &

(Note: The full path name of the executable may not be entered here.) Correspondingly, the user would also check the box named ‘Execute script on Shutdown’, and, to the right of that box, type:

killall -w a2jmidid

However, I have never tried this, as I don’t even have ‘a2jmidid‘ installed.

(Update 5/01/2021, 1h15: )

I must apologize, for the fact that I did not know something at the time I wrote the update from 12/07/2020 above. If the package ‘a2jmidid‘ is in fact installed, on Debian versions ‘Stretch’ and later, starting JACK from QJackCtl will have as effect, automatically to forward all ALSA MIDI Ports to JACK MIDI, without requiring that the user enter any special parameters into the fields ‘Execute script after Startup’ or ‘Execute script on Shutdown’. The fact that this feature automatically gets invoked on JACK startup, at least via the GUI, would actually make it a fatal mistake, to add these lines into the fields as I previously suggested.

Further, when the package is installed and JACK launched, QJackCtl reveals that ALSA MIDI ports will get routed through, even if MIDI devices have been plugged in after JACK start. The following screen-shot shows this in action, as the two entries belonging to my Oxgen-61 Keyboard:

One fact which has already been documented elsewhere is that, when using the package ‘a2jmidid‘, naming inconsistencies will result, for the JACK MIDI ports that are ‘created’. What I found in this one case was that, within Ardour, I was able to choose ‘midi_capture_2′ as input, as shown below (even though the choice was counter-intuitive, according to the QJackCtl GUI as shown before):

And, my Oxygen-61 played the Ardour Instrument for which I had done this. However, even though I saved the session reconfigured in this way, I have no real guarantee, that if I start JACK and then Ardour in the future, and load the same session, Ardour will really be reconnected with the same MIDI device.

Dirk

format: plain


text:
# How to professionally write a constraint, where some variables take certain values of index and some variables should not take some values of index

I am using Pyomo's concrete model to model a MILP mathematical model, coming from multiproduct pipeline scheduling. Some of our continuous variables are: w[i,t], R[i,t], D[i,t]. Fixing the indices i, t, we have the constraint

w[i,t] = w[i,t-1]  +  R[i,t]  -D[i,t]

where, in R[i,t], $$i$$ takes its values from $$I_R \subset I$$ and in D, $$i$$ comes from $$I_D \subset I$$.

Now the problem is that how to model this constraint without conditional statements inside the rule of constraint. As we know, one way is the following code using conditional statement, but I do not want to use it. Is there any other way?

The rule for constraint:

def _cos(i,t):
if i not in I_D and i not in I_R:
return w[i,t] = w[i,t-1]
if i not in I_D and i in I_R:
return w[i,t] = w[i,t-1] +  R[i,t]
if i not in I_R and i in I_D:
return w[i,t] = w[i,t-1] -D[i,t]
if i in I_R and i in I_D:
return w[i,t] = w[i,t-1]  +  R[i,t]  -D[i,t]

• I am not a Pyomo user, but what you are looking for is something like you want to use a specific index, in your case $i$, to represent the outer loop on the two different sets. If so, why not define two different sets/subsets and use two for-loop to do that? Oct 24 at 18:56

I'm assuming your w, R, D variables are not in form of dictionaries (if they are the function below becomes much simpler). I go with "easier to ask for forgiveness than permission" (EAFP) programming.

def _cos(i,t):
try:
_r = R[i,t]
except KeyError:
_r = 0
try:
_d = D[i,t]
except KeyError:
_d = 0
return w[i,t] = w[i,t-1]  +  _r  - _d

• No, they are not dictionaries. Oct 24 at 18:56


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# OOP. Encapsulation and data hiding

Home Forums Programming Software engineering Object-oriented programming OOP. Encapsulation and data hiding

This topic contains 0 replies, has 1 voice, and was last updated by  Васильев Владимир Сергеевич 3 months, 2 weeks ago.

• Author
Posts
• #4079

This article explaines the difference between two close termines used in object-oriented programming: data hiding and encapsulation.

Some people might say it’s an interesting topic to discuss, some might say it’s rather funny, and some might say there is nothing here to discuss. Programming, as other sciences, has its own terminology. Each term describes a certain phenomenon. And those who claim themselves experts in this field, have to know this terminology, whether they like it or not. Therefore, all the attacks and replicas like “Who needs those theory discussions, I’m a programmer, not a teacher,” can be left somewhere else. And if you really think so, you are not a professional. Cause professionals are very accurate and respectful towards what they are using as their professional tools. There is an appropriate phrase which I think can be used in such cases: “It’s a must“. So, here and now we shall talk about termines which many people do not consider to be very simple to understand. What is the difference between Encapsulation and Data Hiding.

In some way, I shall repeat things I have already explained before in the article about Encapsulation. But repeating is another way to learn something. So, let’s start with the Encapsulation.

Those who are familiar with procedural languages such as C, know the problem of global variables. When data is separated from functions which are using them, thus data and logic are defined separately.

Or, for example, C ++, which, though considered to be an object-oriented programming language, yet allows the definition of variables that are stored outside of classes or structures. When I think of what it would look like in the real world, if the laws of encapsulation were violated for some reason, I see a grandmother sitting on the couch, and next to it on the bedside table – her false teeth, a wig, glasses and prosthetic left leg near the couch. A terrible view. Or, for example, I go to a restaurant, order a duck with cherry sauce, and the cook with a shining smile cuts off the duck’s head right in front of my eyes, guts it and throws it into boiling water. Not very enjoyable either.

In the first case – I would like more integrality of the object, in the second – less detail about the process.

This is something encapsulation and data hiding are based on. The details of implementation and the data are packed into an object, and the object is their owner.

On one hand, we don’t need to go into details as to how and what the object performs. On the other hand – it’s easier and safer to operate the object’s data.

Easier – because they are all placed within the object itself; safer – because like this it‘s possible to establish access rules to all of its data and functionality.

And the latter is exactly what is called data hiding.

Data hiding is achieved through access modifiers (in C# those are private, public, protected, protected internal), and each of them defines the type members’ availability.

Now let’s see an example. Here we define the class Person, which encapsulates two private fields – name and age. The modifier private in C# means that access to the field is only possible within the class itself, but not through an instance and not to the derived classes. Access control for reading and writing is provided by the public properties Name and Age. They are verifying assigned values for the range and format.

class Person
{
private string name;
private int age;

public string Name
{
get {return name; }
set
{
if (value.Length> = 2
&& value.Length <= 25)
{
name =
value;
}
else
{

throw new FormatException ("the length of name can't be less than 2 or
longer than 25");
}
}
}

public int Age
{
get {return age; }
set
{
if (value> 1
&& value <120)
{

age = value;
}
else
{

throw new ArgumentOutOfRangeException ();
}
}

}

public override string ToString ()
{
return String.Format ("name - {0},
age - {1}", Name, Age);
}
}

class Program
{
static void Main ()
{
Person person = new Person () {Name
= "John", Age = 147};
Console.WriteLine (person);
}
}

We can say that hiding is one of the the tasks of encapsulation. But it’s not correct to say that encapsulation data hiding are the same thing. Neither it would be right to claim that Encapsulation means only data hiding. Because the aim of Encapsulation lies beyond the scope of data hiding.

If we say that some method encapsulates an algorithm, we mean that this method perfoms a certain calculation, the details of which we don’t have to know, but we can trust its result and can use it. Though we can claim that the object which encapsulates data can hide it – which means it can protect its data from unauthorized access.

Thus, it’s important to distinguish between the concept of encapsulation and data hiding and it’s necessary to put the line between them.

You must be logged in to reply to this topic.


text:
Volume 74 , issue 4 ( 2022 ) back
 NORMALIZED LAPLACIAN ENERGY AND NORMALIZED LAPLACIAN-ENERGY-LIKE INVARIANT OF SOME DERIVED GRAPHS 229$-$241 R. Amin, Sk. Md. Abu Nayeem

Abstract

For a connected graph $G$, the smallest normalized Laplacian eigenvalue is 0 while all others are positive and the largest cannot exceed the value 2. The sum of absolute deviations of the eigenvalues from 1 is called the normalized Laplacian energy, denoted by $\mathbb{LE}(G)$. In analogy with Laplacian-energy-like invariant of $G$, we define here the normalized Laplacian-energy-like as the sum of square roots of normalized Laplacian eigenvalues of $G$, denoted by $\mathbb{LEL}(G)$.

Keywords: Normalized Laplacian energy; normalized Laplacian-energy-like invariant; double graph; extended double cover; Mycielskian.

MSC: 05C50

 ON SOLVABILITY OF QUADRATIC HAMMERSTEIN INTEGRAL EQUATIONS IN HÖLDER SPACES 242$-$248 M. A. Darwish, M. M. A. Metwali, D. O'Regan

Abstract

Using Schauder's fixed point theorem we consider the solvability of a quadratic Hammerstein integral equation in the space of functions satisfying a Hölder condition. An example is included to illustrate our results.

Keywords: Quadratic Hammerstein integral equation; Hölder condition; Schauder fixed point theorem.

MSC: 45G10, 45M99, 47H09

 CANTOR SETS AND FIELDS OF REALS 249$-$259 G. Kuba

Abstract

Our main result is a construction of four families ${\cal C}_1,{\cal C}_2,{\cal B}_1,{\cal B}_2$ which are equipollent with the power set of ${\Bbb R}$ and satisfy the following properties. (i) The members of the families are proper subfields $K$ of ${\Bbb R}$ where ${\Bbb R}$ is algebraic over $K$. (ii) Each field in ${\cal C}_1\cup{\cal C}_2$ contains a {ıt Cantor set}. (iii) Each field in ${\cal B}_1\cup{\cal B}_2$ is a {ıt Bernstein set}. (iv) All fields in ${\cal C}_1\cup{\cal B}_1$ are isomorphic. (v) If $K,L$ are fields in ${\cal C}_2\cup{\cal B}_2$ then $K$ is isomorphic to some subfield of $L$ only in the trivial case $K=L$.

Keywords: Transcendental extensions; descriptive set theory.

MSC: 12F20, 54H05

 NEW FIXED POINT RESULTS FOR ASYMPTOTIC CONTRACTIONS AND ITS APPLICATION TO CANTILEVER BEAM PROBLEMS 260$-$271 S. Karmakar, H. Garai, A. Chanda, L. K. Dey

Abstract

In this article, we deal with some interesting variants of asymptotic contractions, namely Reich type and Chatterjea type weak asymptotic contractions defined on the usual metric spaces. We derive a couple of fixed point results concerning such contractions. Moreover, we look over the existence of solutions to a fourth-order two-point boundary value problem which is a particular type of cantilever beam problems. Furthermore, we construct numerical examples to justify our obtained results.

Keywords: Asymptotic contractions; orbital continuity; cantilever beam problems; boundary value problems.

MSC: 47H10, 54H25

 A GENERALIZATION OF NONSINGULAR REGULAR MAGIC SQUARES 272$-$279 P. Jitjankarn, T. Rungratgasame

Abstract

A generalization of regular magic squares with magic sum $\mu$ is an sq-corner (or square corner) magic square. It is a magic square satisfying the condition that the sum of 4 entries, square symmetrically placed with respect to the center, equals $\frac{4\mu}{n}$. Using the sq-corner magic squares of order $n$, a construction of sq-corner magic squares of order {$n+2$} is derived. Moreover, this construction provides some nonsingular classical sq-corner magic squares of all orders. In particular, a nonsingular regular magic square of any odd order can be constructed under this new method, as well.

Keywords: Nonsingular matrices; determinants; magic squares.

MSC: 15A15, 15F10

 UNIQUENESS OF THE SOLUTION OF A NONLINEAR ALGEBRAIC SYSTEM 280$-$288 P. N. Koumantos

Abstract

In this article we give a sufficient condition for a nonlinear algebraic system of some classes of hypersurfaces to intersect in a unique point and we express the corresponding unique solution in exact form, as well as for the corresponding nonlinear functional system of equations. We conclude extending our results for the functional case in a Banach space of Bochner measurable functions.

Keywords: Nonlinear systems; intersection of hypersurfaces; Bochner integrable functions; equality in the triangle inequality.

MSC: 14P99, 46E30, 26D15, 39B05, 97G70

 NAPOLEON'S THEOREM FROM THE VIEW POINT OF GRÖBNER BASES 289$-$300 M. Čvorak, M. Muzika Dizdarević

Abstract

In this article, we present a new proof of the Napoleon's theorem using algorithmic commutative algebra and algebraic geometry. We also show that, by using the same technique, several related theorems, with the same basic set of objects can be proved. Thus, from the new proof of Napoleon's theorem, we prove the Relative of Napoleon's theorem (result given by B. Grünbaum). Then, we present a new theorem related to Napoleon's theorem. In this theorem the existence of two more quadruplets of equilateral triangles associated with a given triangle was established.

Keywords: Napoleon's theorem; automatic theorem proving; elementary geometry; Gröbner bases.

MSC: 13P10, 68T15, 51M04


text:
# [luatex] Refuses to page break between section headings

Jonas Olson jolson at kth.se
Mon Mar 9 11:54:00 CET 2015

```Den 2015-03-09 10:55, David Carlisle skrev:
> This is unrelated to luatex.

I see. Thank you so much for replying anyway. For the future, what would
be the best place to take issues like this? Is it Stack Exchange?

> latex does not break a page after a section heading.

How come it doesn't even break after the heading when the section is
empty? Is there a switch to change this behaviour?

> If (for some reason) you need a blank document just put some invisible
> paragraph text
> (an empty \mbox{} will do)
>
> so that latex can break after the blank paragraph.

This unfortunately also creates an actual, empty line, which messes up
the spacing. Is there something that takes up zero space, but still
allows for a page break? Otherwise, I suppose I should try to make
something that just looks like a header (bold, large, spacing around it)


text:
# file-local variables vs directory-local variables in Auctex

I have a file with the following content, let's call it tmp.tex

\emph{Hello}

%%% Local Variables:
%%% mode: latex
%%% TeX-master: "main"
%%% End:

and another file main.tex that includes the previous file

\documentclass{article}
\begin{document} \input{tmp} \end{document}

%%% Local Variables:
%%% mode: latex
%%% TeX-master: t
%%% End:

This works. I can open the file tmp.tex and run a compilation command on it. However, I tried moving all my variables to directory level by creating a file .dir-locals.el in the same directory with the following content

((latex-mode
(TeX-master . "main")))

and changed the content of tmp.tex to \emph{Hello} only. Now, if I open the file tmp.tex, the variable TeX-master is not set! Upon closer inspection, the mode that tmp.tex defaults to is plain-tex-mode rather than latex-mode. Of course, I can solve one issue by changing the mode in .dir-locals.el, but it would still mean that the file tmp.tex will not be opened in latex-mode by default.

This is strange, since previously, when I had the comments setting the local variables, the default mode was latex-mode rather than plain-tex-mode.

Is there a way to fix this besides having file-local variables?

• In your file-local approach you are setting the major mode to latex-mode with the mode: latex entry. If you remove that, you will need to use some alternative approach to enforcing the desired mode. i.e. "the default mode was latex-mode rather than plain-tex-mode" is (presumably) not true. – phils Oct 29 '18 at 21:38


text:
# If $$\frac{0.09}{\text{x}}=1$$, then x =

0.09

Explanation

Multiply both sides of the equation by 0.09 to isolate x: 0.09 / x * 0.09 = 1 * 0.09 or x = 0.09. Check by substituting 0.09 for x in the original equation.


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Vol. 6, No. 1, 2012

 Recent Issues
 The Journal Cover Editorial Board Editors' Addresses Editors' Interests About the Journal Scientific Advantages Submission Guidelines Submission Form Subscriptions Editorial Login Contacts Author Index To Appear ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print)
$L$-series of Artin stacks over finite fields

Shenghao Sun

Vol. 6 (2012), No. 1, 47–122
Abstract

We develop the notion of stratifiability in the context of derived categories and the six operations for stacks. Then we reprove the Lefschetz trace formula for stacks, and give the meromorphic continuation of $L$-series (in particular, zeta functions) of ${\mathbb{F}}_{q}$-stacks. We also give an upper bound for the weights of the cohomology groups of stacks, and an “independence of $\ell$” result for a certain class of quotient stacks.

Keywords
$l$-adic cohomology, algebraic stack, Lefschetz trace formula, $L$-function
Mathematical Subject Classification 2010
Primary: 14F20
Secondary: 14F05, 19F27


text:
Elementary Algebra
Introduction to Algebra Linear Equations and Inequalities Functions and Graphs I Lines and thier Graphs Linear Systems Exponents & Polynomials
Intermediate Algebra
Factoring Rational Expressions Rational Equations and Applications Radical Expressions Nonlinear Equations and Applications Functions and Graphs II Exponential and Logarithmic Functions
Precalculus I / College Algebra
Equations and Inequalitites Functions and Graphs Polynomial and Rational Functions Exponential and Logarithmic Functions Systems and Matrices Geometry Basics Conic Sections Sequences and Series
Precalculus II / Trigonometry
The Six Trigonometric Functions Right Triangle Trigonometry Circular Functions Graphs of Trigonometric Functions Trigonometric Identities Trigonometric Equations Oblique Triangles and the Laws Vectors Complex, Parametric, and Polar Forms
Calculus I
Limits and Continuity Derivatives Analysis of Curves Antiderivatives
Calculus II
Transcendental Functions
Geometry Physics Integration Techniques Calculus of Infinity Parametric, Polar, and Conic Curves
Calculus III

Course: College Algebra
Topic: Polynomial and Rational Functions
Subtopic: Polynomial Functions II - Division, Remainder Thm, Factor Thm

Overview

Continuing our study of polynomial functions, in this lesson we concentrate on two important theorems that will enable us to find roots of the functions which in term will enable us to factor the polynomial completely. The ultimate goal is to turn a long polynomial in general form into a product of linear factors.

Objectives

By the end of this topic you should know and be prepared to be tested on:

• 3.3.1 Perform synthetic division
• 3.3.2 Use synthetic division to simplify rational expressions whose denominator is a linear expression of the form (x +- c)
• 3.3.3 Understand and be able to apply the remainder theorem and the factor theorem
• 3.3.4 Use synthetic division to determine if a given number is a zero of the polynomial
• 3.3.5 Know the connection between zeros, roots, and x-intercept points
• 3.3.6 Use theorems and synthetic division to factor a polnomial in the rational realm, real realm, or complex realm
• 3.3.7 Find a polynomial that has given roots

Terminology

Terms you should be able to define: synthetic division, zero, root, remainder theorem, factor theorem, rational vs. real vs. complex realm

Mini-Lectures and Examples

Supplemental Resources (optional)

If you need supplemental tutorial videos with examples relevant to this section go to James Sousa's MathIsPower4U and search for topics:
"Determining the Characteristics of Polynomial Functions"
"Finding the Zeros of Polynomial Functions"

rev. 2020-10-10


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Cokernel

# Cokernel

Discussion

Encyclopedia
In mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the cokernel of a linear mapping of vector spaces f : XY is the quotient space
Quotient space (linear algebra)
In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by "collapsing" N to zero. The space obtained is called a quotient space and is denoted V/N ....

Y/im(f) of the codomain
Codomain
In mathematics, the codomain or target set of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation...

of f by the image of f.

Cokernels are dual
Dual (category theory)
In category theory, a branch of mathematics, duality is a correspondence between properties of a category C and so-called dual properties of the opposite category Cop...

to the kernels of category theory
Kernel (category theory)
In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms, the kernels of module homomorphisms and certain other kernels from algebra...

, hence the name: the kernel is a subobject of the domain (it maps to the domain), while the cokernel is a quotient object of the codomain (it maps from the codomain).

Intuitively, given an equation that one is seeking to solve,
the cokernel measures the constraints that y must satisfy for this equation to have a solution – the obstructions to a solution – while the kernel measures the degrees of freedom in a solution, if one exists. This is elaborated in intuition, below.

More generally, the cokernel of a morphism
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...

f : XY in some category
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...

(e.g. a homomorphism
Group homomorphism
In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...

between group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

s or a bounded linear operator between Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

s) is an object Q and a morphism q : YQ such that the composition q f is the zero morphism
Zero morphism
In category theory, a zero morphism is a special kind of morphism exhibiting properties like those to and from a zero object.Suppose C is a category, and f : X → Y is a morphism in C...

of the category, and furthermore q is universal with respect to this property. Often the map q is understood, and Q itself is called the cokernel of f.

In many situations in abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

, such as for abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

s, vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

s or module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...

s, the cokernel of the homomorphism
Homomorphism
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...

f : XY is the quotient of Y by the image
Image (mathematics)
In mathematics, an image is the subset of a function's codomain which is the output of the function on a subset of its domain. Precisely, evaluating the function at each element of a subset X of the domain produces a set called the image of X under or through the function...

of f. In topological
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

settings, such as with bounded linear operators between Hilbert spaces, one typically has to take the closure
Closure (mathematics)
In mathematics, a set is said to be closed under some operation if performance of that operation on members of the set always produces a unique member of the same set. For example, the real numbers are closed under subtraction, but the natural numbers are not: 3 and 8 are both natural numbers, but...

of the image before passing to the quotient.

## Formal definition

One can define the cokernel in the general framework of category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...

. In order for the definition to make sense the category in question must have zero morphism
Zero morphism
In category theory, a zero morphism is a special kind of morphism exhibiting properties like those to and from a zero object.Suppose C is a category, and f : X → Y is a morphism in C...

s. The cokernel of a morphism
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...

f : XY is defined as the coequalizer
Coequalizer
In category theory, a coequalizer is a generalization of a quotient by an equivalence relation to objects in an arbitrary category...

of f and the zero morphism 0XY : XY.

Explicitly, this means the following. The cokernel of f : XY is an object Q together with a morphism q : YQ such that the diagram
commutes
Commutative diagram
In mathematics, and especially in category theory, a commutative diagram is a diagram of objects and morphisms such that all directed paths in the diagram with the same start and endpoints lead to the same result by composition...

. Moreover the morphism q must be universal
Universal property
In various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem. The definition of a universal property uses the language of category theory to make this notion precise and to study it abstractly.This article gives a general treatment...

for this diagram, i.e. any other such q′: YQ′ can be obtained by composing q with a unique morphism u : QQ′:
As with all universal constructions the cokernel, if it exists, is unique up to
Up to
In mathematics, the phrase "up to x" means "disregarding a possible difference in  x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...

a unique isomorphism
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations.  If there exists an isomorphism between two structures, the two structures are said to be isomorphic.  In a certain sense, isomorphic structures are...

, or more precisely: if q : YQ and q‘ : YQ‘ are two cokernels of f : XY, then there exists a unique isomorphism u : QQ‘ with q‘ = u q.

Like all coequalizers, the cokernel q : YQ is necessarily an epimorphism
Epimorphism
In category theory, an epimorphism is a morphism f : X → Y which is right-cancellative in the sense that, for all morphisms ,...

. Conversely an epimorphism is called normal
Normal morphism
In category theory and its applications to mathematics, a normal monomorphism or conormal epimorphism is a particularly well-behaved type of morphism.A normal category is a category in which every monomorphism is normal...

(or conormal) if it is the cokernel of some morphism. A category is called conormal if every epimorphism is normal (e.g. the category of groups
Category of groups
In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category...

is conormal).

### Examples

In the category of groups
Category of groups
In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category...

, the cokernel of a group homomorphism
Group homomorphism
In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...

f : GH is the quotient
Quotient group
In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...

of H by the normal closure
Normal closure
The term normal closure is used in two senses in mathematics:* In group theory, the normal closure of a subset of a group is the smallest normal subgroup that contains the subset; see conjugate closure....

of the image of f. In the case of abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

s, since every subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...

is normal, the cokernel is just H modulo
Ideal (ring theory)
In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....

the image of f:
coker(f) = H / im(f).

### Special cases

In mathematics, specifically in category theory, a preadditive category is a category that is enriched over the monoidal category of abelian groups...

, it makes sense to add and subtract morphisms. In such a category, the coequalizer
Coequalizer
In category theory, a coequalizer is a generalization of a quotient by an equivalence relation to objects in an arbitrary category...

of two morphisms f and g (if it exists) is just the cokernel of their difference:.

In an abelian category
Abelian category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...

(a special kind of preadditive category) the image
Image (category theory)
Given a category C and a morphismf\colon X\to Y in C, the image of f is a monomorphism h\colon I\to Y satisfying the following universal property:#There exists a morphism g\colon X\to I such that f = hg....

and coimage
Coimage
In algebra, the coimage of a homomorphismis the quotientof domain and kernel.The coimage is canonically isomorphic to the image by the first isomorphism theorem, when that theorem applies....

of a morphism f are given by.

In particular, every abelian category is normal (and conormal as well). That is, every monomorphism
Monomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation X \hookrightarrow Y....

m can be written as the kernel of some morphism. Specifically, m is the kernel of its own cokernel:

## Intuition

The cokernel can be thought of as the space of constraints that an equation must satisfy, as the space of obstructions, just as the kernel is the space of solutions.

Formally, one may connect the kernel and the cokernel by the exact sequence
Exact sequence
An exact sequence is a concept in mathematics, especially in homological algebra and other applications of abelian category theory, as well as in differential geometry and group theory...

These can be interpreted thus: given a linear equation to solve,
• the kernel is the space of solutions to the homogeneous equation and its dimension is the number of degrees of freedom in a solution, if it exists;
• the cokernel is the space of constraints that must be satisfied if the equation is to have a solution, and its dimension is the number of constraints that must be satisfied for the equation to have a solution.

The dimension of the cokernel plus the dimension of the image (the rank) add up to the dimension of the target space, as the dimension of the quotient space is simply the dimension of the space minus the dimension of the image.

As a simple example, consider the map given by
Then for an equation to have a solution, we must have (one constraint), and in that case the solution space is or equivalently stated, (one degree of freedom). The kernel may be expressed as the subspace the value of x is the freedom in a solution – while the cokernel may be expressed via the map given a vector the value of a is the obstruction to there being a solution.


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Home > CFA Level 1 > Bond pricing with spot rates

Bond pricing with spot rates

One method for determining bond price is to discount all bond cash flows at the market discount rate. But since the yield curve is not flat, a more appropriate approach is one where each bond cash flow is discounted separately at the interest rate which corresponds to its maturity, i.e. the relevant spot interest rates.

Spot interest rate

A spot interest rate (also called a zero rate) is the yield to maturity on a zero-coupon bond maturing on the date of each cash flow.

Following is the general expression for bond pricing using spot rates

$P=\frac{PMT}{\left(1+Z_1\right)^1}+\frac{PMT}{\left(1+Z_2\right)^2}+…+\frac{PMT+FV}{\left(1+Z_N\right)^N}$

Where Z1, Z2 and Zn are the relevant spot rates.

Example

A bond has 3-years till maturity, and it pays 4% coupon rate. If the one-year, two-year and three-year spot rates at 3%, 4%, and 5% respectively, the value of the bond would be 97.4208 as shown below:

$P=\frac{4}{\left(1+3\%\right)^1}+\frac{4}{\left(1+4\%\right)^2}+\frac{4+100}{\left(1+5\%\right)^3}=97.4208$

Since the price is less than 100, the bond is trading at a discount. The bond’s yield to maturity must be less than the coupon rate. Using the formula for bond price using a uniform rate, we can find out the bond’s yield to maturity.

$97.4208=\frac{4}{\left(1+r\right)^1}+\frac{4}{\left(1+r\right)^2}+\frac{4+100}{\left(1+r\right)^3}$

It works out to 4.95%.


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# Transfer Learning and Batch Normalization layers

I am trying to use ResNet152 ( pretrained on imagenet) to solve a certain classification task. According to Keras, one strategy for transfer learning is that we do not use the Resnet classifier and then we design and train our own classifier but while freezing the Resnet layers ( phase 1) . After that, we unfreeze the Resnet layers and fine-tune the model as a whole ( phase 2) . To do so we set the trainable attribute of the Resnet model to false and we need to set the training argument to false and for phase 2 we need to set the trainable attribute to true so that we can fine tune the whole layers of the model while keeping the batch normalization layers in inference mode, since we have passed the argument training with False in phase 1 and therefore we will not alter the already learned statistics in the batch normalization layers from imagenet . Now what I did is that I have followed the same steps for phase 1 , however, I have saved the model using model.save() , then later ( after many days) I have loaded again the model and I have then fined tuned this model. So here is my question, when I have loaded the model I have compiled and trained the model directly, I am not sure that if the batch normalization layers will work in inference or training mode ? In other words, after loading the model the training argument still has a False value, or do I need to do some extra step after loading the model to make sure that the Batch normalization layers will work in inference mode?


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0

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% operators that are separated from the operand by a space

% autosize deliminaters

% operators that require brackets

% operators that require parentheses

% Paper specific

Previously, I presented the theory of DRL for finite MDPs. Now, although I believe this theory can be generalized much beyond finite MDPs (at least to finite POMDPs and infinite POMDPs that satisfy some geometric and/or dynamical systems theoretic assumptions), it cannot work for arbitrary environments (without requiring the advisor to be more than "merely sane"). Constructing a counterexample is not difficult, but it seemed worthwhile to write it down.

Let be the set of actions and the set of observations. Define s.t. for any and

That is, is the first time action (stands for "commit") appears in the history . Now, consider the following environments , .

Also, define the reward function by setting for any history that ends with and for any other history. Denote . That is, both universes count the number of actions and until time (the first time action is taken). At times between and , they produce rewards with frequency that equals the relative fraction of the corresponding action in the count. Before and after , they produce no rewards.

Now, we haven't defined sane policies for arbitrary environments, but the spirit of the definition is that a sane policy is unlikely to take actions with major irreversible long-term negative consequences and has to have some non-negligible probability of taking an optimal (or at least nearly optimal) action. For example, we might define it as follows

#Definition

Consider , and a universe . A policy is called -sane for when for any

i.

ii.

Here, we consider the asymptotics , but should include the scenario so that actions with short-term negative consequences are allowed (otherwise this would be some sort of "rationality" requirement, similar to what's needed for DIRL, rather than a mere "sanity" requirement).

The optimal policy for (respectively ) is taking action (resp. ) until some time and taking action at this time (obviously, the following actions don't affect anything). Let be policies that are optimal even off-policy (so that their decision when to take action depends on the history of actions before). It is easy to see that the following policies are -sane for the respective universes and some "legitimate" choice of :

That is, whenever suggests taking action , complies with it, and whenever suggests taking action or , flips a coin to decide which one to take. Indeed, in expectation, taking the wrong action out of loses one time moment of reward and is therefore equivalent to a "short-term" loss.

It is thus impossible for the agent to distinguish between the universes and until action is taken: both the environment and the advisor behave identically until this point. On the other hand, after action is taken it is too late to do anything. Evidently, the hypothesis class is not learnable.

In principle, it might be possible to get around this obstacle by formulating a condition on the advisor in which losses that are minor in size but far away in time are also ruled out. However, such a condition might prove too stringent. Mostly, I hope to ultimately extend the formalism to incomplete models in which case certain restrictions on the type of incomplete model might be acceptable. For example, the incomplete equivalent of an MDP might be a stochastic game against some arbitrary "opponent."

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text:
Research ContributionArnold Mathematical Journal

Received: 16 March 2016 / Revised: 9 December 2016 / Accepted: 23 December 2016

Random Chain Complexes
ginzburg@ucsc.edu,
Department of Mathematics,
UC Santa Cruz,
Santa Cruz CA 95064 USA
The work is partially supported
by NSF grant DMS-1308501
and dmitrii.pasechnik@cs.ox.ac.uk
Department of Computer Science
University of Oxford,
Wolfson Building,
Parks Road Oxford OX1 3QD UK
The work is partially supported
by the EU Horizon 2020
research and innovation programme,
grant agreement OpenDreamKit No. 676541
###### Abstract
We study random, finite-dimensional, ungraded chain complexes over a finite field and show that for a uniformly distributed differential a complex has the smallest possible homology with the highest probability: either zero or one- dimensional homology depending on the parity of the dimension of the complex. We prove that as the order of the field goes to infinity the probability distribution concentrates in the smallest possible dimension of the homology. On the other hand, the limit probability distribution, as the dimension of the complex goes to infinity, is a super-exponentially decreasing, but strictly positive, function of the dimension of the homology.
###### Keywords
Random chain complexes, Homology, Floer theory
###### Mathematics Subject Classification
05E99, 55U15, 53D99, 60D99

## 1. Introduction

We study random, finite-dimensional, ungraded chain complexes over a finite field and we are interested in the probability that such a complex has homology of a given dimension. We show that for a uniformly distributed differential the complex has the smallest possible homology with the highest probability.

To be more specific, consider an $n$-dimensional vector space $V$ over a finite field $\F=\F_q$ of order $q$ and let $D$ be a differential on $V$, i.e., a linear operator $D\colon V\to V$ with $D^2=0$. We are interested in the probability $p_r(q,n)$ with which a chain complex $(V,D)$ has homology $\ker D/\im D$ of a given dimension $r$ for fixed $n$ and $q$. The differential $D$ is uniformly distributed and $p_r(q,n)$ is simply the ratio $c_r(q,n)/c(q,n)$, where $c_r(q,n)$ is the number of complexes with $r$-dimensional homology and $c(q,n)$ is the number of all complexes. The numbers $c_r(q,n)$ are explicitly calculated by [Kovacs1987]; see also Theorem 2.2.

We mainly focus on large complexes, i.e., on the limits as $q$ or $n$ go to infinity. Clearly, $r$ and $n$ must have the same parity and we separately analyze the asymptotic behavior of the sequence $p_0(q,n), p_2(q,n),\ldots$, where $n$ is even, and the sequence $p_1(q,n), p_3(q,n),\ldots$ for $n$ odd.

As $q\to\infty$ with $n$ fixed, the probability concentrates in the lowest possible dimension, i.e., $p_0(q,n)\to 1$ or $p_1(q,n)\to 1$ depending on the parity of $n$, while $p_r(q,n)\to 0$ for $r>1$. This is consistent with the observation that over $\C$ and even $\R$ (see Lemma 3.1) a generic complex has $0$- or $1$-dimensional homology, i.e., that such complexes form the highest dimensional stratum in the variety of all $n$-dimensional complexes. Indeed, one can expect the probability distributions for large $q$ to approximate the generic situation in zero characteristic. We do not know, however, if the density functions converge in any sense as $q\to\infty$ to some probability density on the variety of $n$-dimensional complexes over, e.g., $\R$.

When $q$ and $r$ are fixed and $n\to\infty$ through either even or odd integers depending on the parity of $r$, the situation is more subtle. In this case, all limit probabilities $p_r(q)=\lim_{n\to\infty}p_r(q,n)$ are positive. However, the sequences $p_0,p_2,\ldots$ and $p_1,p_3,\ldots$ are super-exponentially decreasing and for a large $q$ all terms in these sequences but the first one are very close to zero while the first is then, of course, close to 1. When $q=2$ and $r$ is even, we have $p_0\approx 0.6$, $p_2\approx 0.4$, $p_4\approx 0.0075$ and other terms are very small. We explicitly calculate the ratios $p_r(q)/p_0(q)$ and $p_r(q)/p_1(q)$ and $p_0$ and $p_1$ in Theorem 2.1.

The proofs of these facts are elementary and quite simple. However, we have not been able to find in the literature any probability calculations in this basic case where random chain complexes are stripped of all additional structures including a grading. (The combinatorial part of our proof, Theorem 2.2, is contained in [[Kovacs1987], Lemma 5].) In contrast, random complexes of geometrical origin and underlying random geometrical and topological objects have been studied extensively and from various perspectives. Among such random objects are, for instance, random simplicial complexes of various types [see ([Aronshtam et al.2013],[Bobrowski and Kahle2014],[Costa and Farber2014],[Costa and Farber2015],[Kahle2011],[Meshulam2013],[Meshulam and Wallach2009],[Pippenger and Schleich2006],[Yogeshwaran et al.2014]) and references therein] and random Morse functions [see, e.g., ([Arnold2006],[Arnold2007],[Collier et al.2017],[Nicolaescu2012])].

These works utilize several models of randomness all of which appear to be quite different from the one, admittedly rather naive, used here. This makes a direct comparison difficult. One way to interpret our result is that, for a large complex, sufficiently non-trivial homology is indicative of some structure, a constraint limiting randomness. Note that such a structure can be as simple as a $\Z$-grading confined to a fixed range of degrees. A dimensional constraint of this type is usually inherent in geometrical complexes, and it would be interesting to analyze its effect (if any) on the probability distribution in our purely algebraic setting. Another consequence of the result is that the assertion that a complex has large homology carries more information than the assertion that it has small homology.

The main motivation for our setting comes from Hamiltonian Floer theory for closed symplectic manifolds; see, e.g., [Salamon1999] and references therein. A Hamiltonian diffeomorphism is the time-one map of the isotopy generated by a time-dependent Hamiltonian. To such a diffeomorphism one can associate a certain complex, called the Floer complex, generated by its fixed points or, equivalently, the one-periodic orbits of the isotopy. Hence the dimension of the Floer homology gives a lower bound for the number of one-periodic orbits. The homology is independent of the Hamiltonian diffeomorphism. In addition, one can fix the free homotopy class of the orbits. (This construction is similar to Morse theory and, in fact, Floer theory is a version of Morse theory for the action functional.)

In many instances, e.g., often generically or for all symplectic manifolds with vanishing first Chern class such as tori, the dimension of the Floer complex grows with the order of iteration of the diffeomorphism; see [Ginzburg and G[U+00A8]urel2015]. In other words, the complex gets larger and larger as time in this discrete dynamical system grows. Moreover, the differential in the complex is usually impossible to describe explicitly, and hence it makes sense to compare the behavior of the complex and its homology with the generic or random situation. The Floer homology for contractible periodic orbits is isomorphic to the homology of the underlying manifold. Therefore, by our result, even though the Floer complex appears to be very "noisy" for large iterations and random on a bounded action scale, it has large homology groups and is actually very far from random. For non-contractible orbits, the dimension of the Floer complex is also known to grow in many settings; see [Ginzburg and G[U+00A8]urel2015],[G[U+00A8]urel2013]. However, in this case the Floer homology is zero and the complex may well be close to random. Note also that in some instances the Floer complex is $\Z$-graded, but the grading is not supported within any specific interval of degrees. Moreover, in contrast with geometrical random complexes, the grading range of the Floer homology usually grows with the order of iteration ([Salamon and Zehnder1992]), and while it is not clear how to correctly account for an unbounded grading in a random model, such a grading is unlikely to affect the probability distribution.

One aspect of Floer theory which is completely ignored in our model is the action filtration. This filtration is extremely important and, in particular, it allows one to treat Floer theory in the context of persistent homology and topological data analysis; see [Carlsson2009],[Ghrist2008]. This connection has recently been explored in [Polterovich and Shelukhin2014],[Usher and Zhang2015]. However, it is not entirely clear how to meaningfully incorporate the action filtration into our model.

## 2. Main Results

Let, as in the introduction, $(V,D)$ be an ungraded $n$-dimensional chain complex with differential $D$ over a finite field $\F=\F_q$ of order $q$. In other words, $V=\F^n$ and $D$ is a linear operator on $V$ with $D^2=0$. We denote by $c(q,n)$ the number of such complexes, i.e., the number of differentials $D$. The dimension $r$ of the homology $\ker D/\im D$ has the same parity as $n$ and we let $c_r(q,n)$ be the number of complexes with homology of dimension $r$. (In what follows, we always assume that $r$ and $n$ have the same parity.) Clearly, \begin{eqnarray*} c(q,n)=c_0(q,n)+c_2(q,n)+\cdots+c_{n}(q,n) \end{eqnarray*} when $n$ is even and \begin{eqnarray*} c(q,n)=c_1(q,n)+c_3(q,n)+\cdots+c_{n}(q,n) \end{eqnarray*} when $n$ is odd.

Furthermore, denote by \begin{eqnarray*} p_r(q,n)=\frac{c_r(q,n)}{c(q,n)} \end{eqnarray*} the probability (with respect to the uniform distribution) of a complex to have $r$-dimensional homology. Our main result describes the behavior of $p_r(q,n)$ as the size of the complex, i.e., $q$ or $n$, goes to infinity.

###### Theorem 2.1.
Let $p_r(q,n)$ be as above.
1. (i) For a fixed $n$, we have \begin{equation*} \lim_{q\to\infty} p_r(q,n)=0\textrm{ when } r>1, \end{equation*} and $p_0(q,n)\to 1$ when $n$ is even and $p_1(q,n)\to 1$ when $n$ is odd as $q\to\infty$.
2. (ii) For a fixed $q$ and $r$, the limits \begin{eqnarray*} p_r(q)=\lim_{n\to\infty} p_r(q,n) \end{eqnarray*} exist and $0  $$\label{eq:lim} \frac{p_r(q)}{p_0(q)}=\frac{q^{r/2}}{\prod\nolimits_{j=1}^r(q^j-1)}$$ (2.1) and  $$\label{eq:S} p_0(q)=\frac{1}{1+S},\quad \textrm{ where }\quad S=\sum_{k=1}^\infty \frac{q^{k}}{\prod\nolimits_{j=1}^{2k}(q^j-1)}.$$ (2.2) When$ r\geq 3$is odd, \begin{eqnarray*} \frac{p_r(q)}{p_1(q)}=\frac{(q-1)q^{(r-1)/2}}{\prod\nolimits_{j=1}^r(q^j-1)} \end{eqnarray*} and \begin{eqnarray*} p_1(q)=\frac{1}{1+S'},\quad \textrm{ where }\quad S'=(q-1)\sum_{k=1}^\infty \frac{q^{k}}{\prod\nolimits_{j=1}^{2k+1}(q^j-1)}. \end{eqnarray*} The proof of this theorem is based on an explicit calculation of$ c_r(q,n)$. To state the result, denote by$ \GL_k(q)$the general linear group of$ k\times k$invertible matrices over$ \F_q$and recall that \begin{eqnarray*} |\GL_k(q)|=q^{k(k-1)/2}\prod_{j=1}^k(q^j-1). \end{eqnarray*} Then we have the following particular case of [[Kovacs1987], Lemma 5]. ###### Theorem 2.2. ([Kovacs1987]) Let as above$ c_r(q,n)$be the number of$ n$-dimensional complexes over$ \F_q$with homology of dimension$ r$. Then  $$\label{eq:number} c_r(q,n)=\frac{|\GL_n(q)|}{|\GL_m(q)|\cdot |\GL_r(q)|\cdot q^{2mr+m^2}}\, ,$$ (2.3) where$ 2m+r=n$. Even though this result is not new, for the sake of completeness we include its proof, which is very simple and short, in the next section. ###### Remark 2.3. We do not have simple expressions for the probabilities$ p_r(q,n)$and the total number of complexes$ c(q,n)$. However, when$ q=2$, the differentials$ D$are in one-to-one correspondence with involutions of$ \F_2^n$. (An involution necessarily has the form$ I+D$and, as is easy to see, different differentials$ D$give rise to different involutions.) Hence,$ c(2,n)$is equal to the number of involutions. This number is expressed in [Fulman and Vinroot2014] via a generating function and an asymptotic formula for$ c(q,n)$has been recently obtained in [Fulman et al.2016]. It is possible that at least when$ q=2$our probability formulas can be further simplified using the results from those papers. ## 3. Proofs The proof of Theorem 2.2 is based on the observation that the differential in a finite-dimensional complex over any field$ \F$can be brought to its Jordan normal form or, equivalently, a complex over$ \F$can be decomposed into a sum of elementary complexes, i.e., into a sum of two-dimensional complexes with zero homology and one-dimensional complexes. To be more precise, we have the following elementary observation. ###### Lemma 3.1. Let$ V$be a finite-dimensional vector space over an arbitrary field$ \F$and let$ D\colon V\to V$be an operator with$ D^2=0$. Then, in some basis,$ D$can be written as a direct sum of$ 1\times 1$and$ 2\times 2$Jordan blocks with zero eigenvalues. When$ \F$is algebraically closed, this follows immediately from the Jordan normal form theorem. Hence, the emphasis here is on the fact that the field$ \F$is immaterial. For the sake of completeness, we outline a proof of the lemma. ###### Proof. Let us pick an arbitrary basis$ \{e_1,\ldots, e_m\}$of$ \im D$and extend it to a basis of$ \ker D\supset \im D$by adding elements$ \{f_1,\ldots,f_r\}$. Furthermore, pick arbitrary vectors$ e'_i$with$ De'_i=e_i$. Then$ \{e'_1, e_1, \ldots, e'_m, e_m, f_1, \ldots, f_r\}$is the required basis of$ V$. ###### Proof. Proof of Theorem 2.2 Let$ D$be a differential on an$ n$-dimensional vector space$ V$over a finite field$ \F=\F_q$. Assume that the homology of the complex$ (V,D)$is$ r$-dimensional. By Lemma 3.1,$ D$is conjugate to the map$ D_r$which is the direct sum of$ r 1\times 1$zero blocks and$ m 2\times 2$Jordan blocks with zero eigenvalues, where$ 2m+r=n$. Let$ C_r$be the centralizer of$ D_r$in$ \GL_n(q)$. The complexes with$ r$-dimensional homology are in one-to-one correspondence with$ \GL_n(q)/C_r$. Thus, to prove (2.3) , it suffices to show that  $$\label{eq:center} |C_r|=|\GL_m(q)|\cdot |\GL_r(q)|\cdot q^{2mr+m^2} .$$ (3.1) The elements of$ C_r$are$ n\times n$invertible matrices$ X\in \GL_n(q)$commuting with$ D_r$. In what follows, it is convenient to work with the basis$ e_1, \ldots, e_m, f_1, \ldots, f_r, e'_1,\ldots,e'_m$in the notation from the proof of Lemma 3.1. Thus we can think of$ X$as a$ 3\times 3$-block matrix with$ m\times m$block$ X_{11}$, the block$ X_{12}$having size$ m\times r$, and$ X_{13}$being again$ m\times m$, etc. In the same format,$ D_r$is then the matrix with only one non-zero block. This is the top-right corner$ m\times m$-block, which is$ I$. Then, as a straightforward calculation shows, the commutation relation$ XD_r=D_rX$amounts to the conditions that$ X_{11}=X_{33}$, and$ X_{21}=0$,$ X_{31}=0$and$ X_{32}=0$. In particular,$ X$is an upper block-triangular matrix. Hence,$ X$is invertible if and only if$ X_{11}=X_{33}$and$ X_{22}$are invertible. There are no constraints on the remaining blocks$ X_{12}$,$ X_{13}$and$ X_{23}$. Now (3.1) follows. ###### Proof of Theorem 2.1. Throughout the proof, we assume that$ r$and$ n$are even. The case where these parameters are odd can be handled in a similar fashion. As the first step, we express$ p_r(q,n)/p_0(q,n)explicitly. Clearly, \begin{eqnarray*} \frac{p_r(q,n)}{p_0(q,n)}=\frac{c_r(q,n)}{c_0(q,n)}=\frac{|C_0|}{|C_r|}. \end{eqnarray*} Using (2.3) or (3.1) and tidying up the resulting expression, we have \begin{align*} \frac{p_r(q,n)}{p_0(q,n)} &=\frac{q^{n^2/4}\cdot q^{n(n/2-1)/4}\cdot \prod_{j=1}^{n/2}(q^j-1)} {q^{2mr+m^2}\cdot q^{m(m-1)/2}\cdot q^{r(r-1)/2}\cdot\prod\nolimits_{j=1}^{r}(q^j-1) \cdot\prod\nolimits_{j=1}^{m}(q^j-1)} \\ &=\frac{\prod\nolimits_{j=m+1}^{m+r/2}(q^j-1)} {q^{mr/2+r(r/2-1)/4}\cdot \prod\nolimits_{j=1}^{r}(q^j-1) }, \end{align*} where as above n=2m+r$and$ r\geq 2$, which we can then rewrite as  $$\label{eq:ratio} \frac{p_r(q,n)}{p_0(q,n)}=\frac{q^{r/2}} {\prod\nolimits_{j=1}^{r}(q^j-1) }\cdot\prod\limits_{j=1}^{r/2}\left(1-\frac{1}{q^{m+j}}\right).$$ (3.2) Now it is clear that \begin{eqnarray*} \frac{p_r(q,n)}{p_0(q,n)}\sim q^{-r^2/2}\quad\textrm{ as }\quad q\to\infty \end{eqnarray*} with$ r\geq 2$and$ n$fixed. In particular, this ratio goes to zero as$ q\to \infty$. The number of the terms in the sum \begin{eqnarray*} \sum_j p_j(q,n)=1 \end{eqnarray*} with$ j$ranging through even integers from$ 0$to$ n$is equal to$ n/2+1$and thus this number is independent of$ q$. Hence,$ p_0(q,n)\to 1$and$ p_r(q,n)\to 0$when$ r\geq 2$as$ q\to\infty$. This proves the first assertion of the theorem. To prove the second part, first note that by (3.2)  $$\label{eq:limspsratio} \frac{p_r(q,n)}{p_0(q,n)}\to \frac{q^{r/2}}{\prod\nolimits_{j=1}^r(q^j-1)}$$ (3.3) as$ m\to \infty$or, equivalently,$ n\to \infty$with$ r$and$ q$fixed. Furthermore, in a similar vein, it is not hard to show that \begin{eqnarray*} \sum_{r>0} \frac{p_r(q,n)}{p_0(q,n)}\to S:=\sum_r \frac{q^{r/2}}{\prod\nolimits_{j=1}^r(q^j-1)}\quad\textrm{ as } n\to\infty, \end{eqnarray*} where, on the left, the sum is taken over all even integers from$ 2$to$ n$and, on the right, the sum is over all even integers$ r\geq 2$. Therefore, letting$ n\to\infty$in the identity \begin{eqnarray*} 1+\sum_{r>0} \frac{p_r(q,n)}{p_0(q,n)}=\frac{1}{p_0(q,n)}, \end{eqnarray*} we conclude that the limit$ p_0(q)=\lim_{n\to\infty}p_0(q,n)$exists and$ p_0(q)=1/(1+S)$, which proves (2.2) . Now, by (3.3) , the limits$ p_r(q)=\lim_{n\to\infty}p_r(q,n)$for$ r\geq 2$also exist, and hence (2.1) holds. This completes the proof of the theorem. ###### Remark 3.2. The sequence$ p_r(q,n)$is decreasing as a function of$ r$. This readily follows from (3.2) . ###### Acknowledgements. The authors are grateful to Robert Ghrist, Ba[U+00B8]sak G[U+00A8]urel, Jiang-Hua Lu, Roy Meshulam and Leonid Polterovich for useful discussions and comments. The authors would also like to thank the referee for pointing out [[Kovacs1987], Lemma 5] to them. 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# American Institute of Mathematical Sciences

March  2015, 4(1): 21-38. doi: 10.3934/eect.2015.4.21

## Optimal energy decay rate of Rayleigh beam equation with only one boundary control force

 1 Université Libanaise, EDST, Equipe EDP-AN, Hadath, Beyrouth, Lebanon 2 Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, Institut des Sciences et Techniques of Valenciennes, F-59313 - Valenciennes Cedex 9 3 Université Libanaise, Sciences 1 et EDST, Equipe EDP-AN, Hadath, Beyrouth, Lebanon

Received  July 2014 Revised  January 2015 Published  February 2015

We consider a clamped Rayleigh beam equation subject to only one boundary control force. Using an explicit approximation, we first give the asymptotic expansion of eigenvalues and eigenfunctions of the undamped underlying system. We next establish a polynomial energy decay rate for smooth initial data via an observability inequality of the corresponding undamped problem combined with a boundedness property of the transfer function of the associated undamped problem. Finally, by a frequency domain approach, using the real part of the asymptotic expansion of eigenvalues of the infinitesimal generator of the associated semigroup, we prove that the obtained energy decay rate is optimal.
Citation: Maya Bassam, Denis Mercier, Ali Wehbe. Optimal energy decay rate of Rayleigh beam equation with only one boundary control force. Evolution Equations & Control Theory, 2015, 4 (1) : 21-38. doi: 10.3934/eect.2015.4.21
##### References:
 [1] K. Ammari and M. Tucsnak, Stabilizaton of second order evolution equations by class of unbounded feedbacks,, ESIAM, 6 (2001), 361. doi: 10.1051/cocv:2001114. [2] K. Ammari and M. Tucsnak, Stabilization of Bernoulli-Euler beams by means of a pointwise feedback force,, SIAM, 39 (2000), 1160. doi: 10.1137/S0363012998349315. [3] C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces,, J. Evol. Equ., 8 (2008), 765. doi: 10.1007/s00028-008-0424-1. [4] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups,, Math. Ann., 347 (2010), 455. doi: 10.1007/s00208-009-0439-0. [5] H. Brezis, Analyse Fonctionelle, Théorie et Applications,, Masson, (1983). [6] G. Chen, M. C. Delfour, A. M. Krall and G. Payre, Modeling, stabilization and control of serially connected beams,, SIAM J. Control Optim., 25 (1987), 526. doi: 10.1137/0325029. [7] G. Chen, S. G. Krantz, D. W. Ma, C. E. Wayne and H. H. West, The Euler-Bernoulli beam equation with boundary energy dissipation,, in operator Methods for Optimal Control Problems (ed. Sung J. Lee), (1987), 67. [8] A. E. Ingham, Some trigonometrical inequalities with applications in the theory of series,, Math. Z., 41 (1936), 367. doi: 10.1007/BF01180426. [9] T. Kato, Perturbaton Theory for Linear Operators,, Springer-Verlag, (1976). [10] J. S. Gibson, A note on stabilization of infinite-dimensional linear oscillators by compact linear feedback,, SIAM J. Control Optim., 18 (1980), 311. doi: 10.1137/0318022. [11] B. Z. Guo, J. M. Wang and C. L. Zhou, On the dynamic behavior and stability of controlled connected Rayleigh beams under pointwise output feedback,, ESAIM, 14 (2008), 632. doi: 10.1051/cocv:2008001. [12] J. E. Lagnese, Boundary Stabilization of Thin Plates,, SIAM Studies in Applied Mathematics, (1989). doi: 10.1137/1.9781611970821. [13] Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation,, Z. Angew. Math. Phys., 56 (2005), 630. doi: 10.1007/s00033-004-3073-4. [14] D. Mercier and V. Régnier, Exponential stability of a network of serially connected Euler-Bernoulli beams,, International Journal of Control, 87 (2014), 1266. doi: 10.1080/00207179.2013.874597. [15] Ö. Morgül, Dynamic boundary control of an Euler-Bernoulli beam,, IEEE Trans. Automat. Control, 37 (1992), 639. doi: 10.1109/9.135504. [16] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equation,, Applied Mathematical Sciences, (1983). doi: 10.1007/978-1-4612-5561-1. [17] B. Rao, A Compact perturbation method for the boundary stabilization of the Rayleigh beam equation,, Appl. Math. Optim., 33 (1996), 253. doi: 10.1007/BF01204704. [18] B. Rao, Uniform stabilisation of a hybrid system of elasticity,, SIAM J. Control Optim., 33 (1995), 440. doi: 10.1137/S0363012992239879. [19] B. Rao, Optimal energy decay rate in a damped Rayleigh beam,, Discrete Contin. Dynam. Systems, 4 (1998), 721. doi: 10.3934/dcds.1998.4.721. [20] D. L. Russell, On the mathematical models for the elastic beam with frequence-proportional damping,, in Control and Estimation in Distributed Parameters Systems (ed. H. T. Bank), (1992), 125. doi: 10.1137/1.9781611970982.ch4. [21] J. M. Wang, G. Q. Xu and S. P. Yung, Exponential stability of variable coefcients Rayleigh beams under boundary feedback controls: A Riesz basis approach,, Systems and Control Letters, 51 (2004), 33. doi: 10.1016/S0167-6911(03)00205-6. [22] A. Wehbe, Optimal energy decay rate in the Rayleigh beam equation with boundary dynamical controls,, Bull. Belg. Math. Soc., 13 (2006), 385.

show all references

##### References:
 [1] K. Ammari and M. Tucsnak, Stabilizaton of second order evolution equations by class of unbounded feedbacks,, ESIAM, 6 (2001), 361. doi: 10.1051/cocv:2001114. [2] K. Ammari and M. Tucsnak, Stabilization of Bernoulli-Euler beams by means of a pointwise feedback force,, SIAM, 39 (2000), 1160. doi: 10.1137/S0363012998349315. [3] C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces,, J. Evol. Equ., 8 (2008), 765. doi: 10.1007/s00028-008-0424-1. [4] A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups,, Math. Ann., 347 (2010), 455. doi: 10.1007/s00208-009-0439-0. [5] H. Brezis, Analyse Fonctionelle, Théorie et Applications,, Masson, (1983). [6] G. Chen, M. C. Delfour, A. M. Krall and G. Payre, Modeling, stabilization and control of serially connected beams,, SIAM J. Control Optim., 25 (1987), 526. doi: 10.1137/0325029. [7] G. Chen, S. G. Krantz, D. W. Ma, C. E. Wayne and H. H. West, The Euler-Bernoulli beam equation with boundary energy dissipation,, in operator Methods for Optimal Control Problems (ed. Sung J. Lee), (1987), 67. [8] A. E. Ingham, Some trigonometrical inequalities with applications in the theory of series,, Math. Z., 41 (1936), 367. doi: 10.1007/BF01180426. [9] T. Kato, Perturbaton Theory for Linear Operators,, Springer-Verlag, (1976). [10] J. S. Gibson, A note on stabilization of infinite-dimensional linear oscillators by compact linear feedback,, SIAM J. Control Optim., 18 (1980), 311. doi: 10.1137/0318022. [11] B. Z. Guo, J. M. Wang and C. L. Zhou, On the dynamic behavior and stability of controlled connected Rayleigh beams under pointwise output feedback,, ESAIM, 14 (2008), 632. doi: 10.1051/cocv:2008001. [12] J. E. Lagnese, Boundary Stabilization of Thin Plates,, SIAM Studies in Applied Mathematics, (1989). doi: 10.1137/1.9781611970821. [13] Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation,, Z. Angew. Math. Phys., 56 (2005), 630. doi: 10.1007/s00033-004-3073-4. [14] D. Mercier and V. Régnier, Exponential stability of a network of serially connected Euler-Bernoulli beams,, International Journal of Control, 87 (2014), 1266. doi: 10.1080/00207179.2013.874597. [15] Ö. Morgül, Dynamic boundary control of an Euler-Bernoulli beam,, IEEE Trans. Automat. Control, 37 (1992), 639. doi: 10.1109/9.135504. [16] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equation,, Applied Mathematical Sciences, (1983). doi: 10.1007/978-1-4612-5561-1. [17] B. Rao, A Compact perturbation method for the boundary stabilization of the Rayleigh beam equation,, Appl. Math. Optim., 33 (1996), 253. doi: 10.1007/BF01204704. [18] B. Rao, Uniform stabilisation of a hybrid system of elasticity,, SIAM J. Control Optim., 33 (1995), 440. doi: 10.1137/S0363012992239879. [19] B. Rao, Optimal energy decay rate in a damped Rayleigh beam,, Discrete Contin. Dynam. Systems, 4 (1998), 721. doi: 10.3934/dcds.1998.4.721. [20] D. L. Russell, On the mathematical models for the elastic beam with frequence-proportional damping,, in Control and Estimation in Distributed Parameters Systems (ed. H. T. Bank), (1992), 125. doi: 10.1137/1.9781611970982.ch4. [21] J. M. Wang, G. Q. Xu and S. P. Yung, Exponential stability of variable coefcients Rayleigh beams under boundary feedback controls: A Riesz basis approach,, Systems and Control Letters, 51 (2004), 33. doi: 10.1016/S0167-6911(03)00205-6. [22] A. Wehbe, Optimal energy decay rate in the Rayleigh beam equation with boundary dynamical controls,, Bull. Belg. Math. Soc., 13 (2006), 385.
 [1] Bopeng Rao. Optimal energy decay rate in a damped Rayleigh beam. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 721-734. doi: 10.3934/dcds.1998.4.721 [2] Arnaud Münch, Ademir Fernando Pazoto. Boundary stabilization of a nonlinear shallow beam: theory and numerical approximation. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 197-219. doi: 10.3934/dcdsb.2008.10.197 [3] Moez Daoulatli, Irena Lasiecka, Daniel Toundykov. Uniform energy decay for a wave equation with partially supported nonlinear boundary dissipation without growth restrictions. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 67-94. doi: 10.3934/dcdss.2009.2.67 [4] Jitka Machalová, Horymír Netuka. Optimal control of system governed by the Gao beam equation. Conference Publications, 2015, 2015 (special) : 783-792. doi: 10.3934/proc.2015.0783 [5] Rachid Assel, Mohamed Ghazel. Energy decay for the damped wave equation on an unbounded network. Evolution Equations & Control Theory, 2018, 7 (3) : 335-351. doi: 10.3934/eect.2018017 [6] Kim Dang Phung. Boundary stabilization for the wave equation in a bounded cylindrical domain. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 1057-1093. doi: 10.3934/dcds.2008.20.1057 [7] Moez Daoulatli. Energy decay rates for solutions of the wave equation with linear damping in exterior domain. Evolution Equations & Control Theory, 2016, 5 (1) : 37-59. doi: 10.3934/eect.2016.5.37 [8] Mohammad A. Rammaha, Daniel Toundykov, Zahava Wilstein. Global existence and decay of energy for a nonlinear wave equation with $p$-Laplacian damping. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4361-4390. doi: 10.3934/dcds.2012.32.4361 [9] Yanfang Li, Zhuangyi Liu, Yang Wang. Weak stability of a laminated beam. Mathematical Control & Related Fields, 2018, 8 (3&4) : 789-808. doi: 10.3934/mcrf.2018035 [10] Kaïs Ammari, Mohamed Jellouli, Michel Mehrenberger. Feedback stabilization of a coupled string-beam system. Networks & Heterogeneous Media, 2009, 4 (1) : 19-34. doi: 10.3934/nhm.2009.4.19 [11] Xiu-Fang Liu, Gen-Qi Xu. Exponential stabilization of Timoshenko beam with input and output delays. Mathematical Control & Related Fields, 2016, 6 (2) : 271-292. doi: 10.3934/mcrf.2016004 [12] Ionuţ Munteanu. Exponential stabilization of the stochastic Burgers equation by boundary proportional feedback. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2173-2185. doi: 10.3934/dcds.2019091 [13] Abbes Benaissa, Abderrahmane Kasmi. Well-posedeness and energy decay of solutions to a bresse system with a boundary dissipation of fractional derivative type. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4361-4395. doi: 10.3934/dcdsb.2018168 [14] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao. Polynomial upper bounds for the instability of the nonlinear Schrödinger equation below the energy norm. Communications on Pure & Applied Analysis, 2003, 2 (1) : 33-50. doi: 10.3934/cpaa.2003.2.33 [15] Tae Gab Ha. Global existence and general decay estimates for the viscoelastic equation with acoustic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6899-6919. doi: 10.3934/dcds.2016100 [16] Le Thi Phuong Ngoc, Nguyen Thanh Long. Existence and exponential decay for a nonlinear wave equation with nonlocal boundary conditions. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2001-2029. doi: 10.3934/cpaa.2013.12.2001 [17] Aslihan Demirkaya, Milena Stanislavova. Numerical results on existence and stability of standing and traveling waves for the fourth order beam equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 197-209. doi: 10.3934/dcdsb.2018097 [18] Ferhat Mohamed, Hakem Ali. Energy decay of solutions for the wave equation with a time-varying delay term in the weakly nonlinear internal feedbacks. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 491-506. doi: 10.3934/dcdsb.2017024 [19] Darya V. Verveyko, Andrey Yu. Verisokin. Application of He's method to the modified Rayleigh equation. Conference Publications, 2011, 2011 (Special) : 1423-1431. doi: 10.3934/proc.2011.2011.1423 [20] Kais Ammari, Eduard Feireisl, Serge Nicaise. Polynomial stabilization of some dissipative hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4371-4388. doi: 10.3934/dcds.2014.34.4371

2017 Impact Factor: 1.049


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# 25.5.3 How to Manage Multiple .OMC Files

The configuration of the customized menus can be saved as an .omc file, which can be easily shared with your others. The .omc files should be placed in the User Files Folder, the Group Folder and the Installation Folder of Origin. By Default, all of the .omc files in these folders will be automatically listed under the Preference: Menu. Origin will first list the .omc files under the User Files Folder, then the Group Folder and then the Installation Folder of Origin. The order of the files in the same folder is determined by the alphabetical order of the file names.

The omc.ini file can help you manage the .omc files easily. With the omc.ini file, you can

• Set the maximum number of the files which will be listed under Preference: Menu.
• Change the display order of these files under Preference: Menu.
• Set an .omc file as the default option under Preference: Menu.
 Note: The omc.ini can be saved in the User Files Folder or the Group Folder or the Installation Folder of Origin. And it can only manage the .omc files in the same folder.

## The organization of an omc.ini file

An omc.ini file should contain two sections. The first section is called Conifg, which can be used to set the maximum number of files in a folder that will be listed under Preference: Menu and also the display order of the files. The other section is called Options, which can be used to set an .omc file as the default menu option under Preference: Menu.

Suppose we have four .omc files ImageProcessing.omc, Curve Fititng.omc, Statistics.omc and SignalProcessing.omc in the User Files Folder. And we only want the first three files to be shown. The desired display order is Statistics.omc, ImageProcessing.omc and CurveFitting.omc. Also, the ImageProcessing.omc should be set as default. The following omc.ini file should be:

[Config]
NumOMCLevels=3
;Menu will ordered as the file list below
OMCFile1=Statistics
OMCFile2=ImageProcessing
OMCFile3=CurveFitting

[Options]
OMCLevel=ImageProcessing


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# greek accents with TeX Gyre Termes in lualatex and xelatex

\documentclass[11pt,a4paper]{scrartcl}

\usepackage{fontspec}
\setmainfont[Ligatures=TeX]{TeX Gyre Termes}

\begin{document}
This is English with some Greek: γέγονεν.
Isn't that quite nice? εὕρηκα!
\end{document}

Output using XeLaTeX:

This is English with some Greek: γέγονεν. Isn’t that quite nice? ε�ρηκα!

Output using LuaLaTeX:

This is English with some Greek: γγονεν. Isn’t that quite nice? ερηκα!

It seems that ὕ is missing in XeLaTeX (presumably because it is not part of the font), but έ is present. Both are missing in LuaLaTeX. Any suggestion about how to get LuaLaTeX to find the έ?

• The font doesn't have the character έ (U+03AD). XeLaTeX fakes it somehow, but I don't know how. – Sverre Sep 19 '15 at 15:12
• The faked έ is quite nice looking, it's a pity it can't manage the ὕ in a similar manner! – Joe Corneli Sep 19 '15 at 20:45

As Sverre says, the basic problem here is that the font does not support Greek. Moreover, you are not loading any package to actually deal with multilingual typesetting i.e. either Babel or Polyglossia.

For example, the following works for LuaLaTeX:

\documentclass[11pt,a4paper]{scrartcl}
\usepackage{polyglossia}
\setmainlanguage[variant=british]{english}
\setotherlanguage{greek}
\usepackage{fontspec}
\setmainfont{GFS Artemisia}[Ligatures=TeX]
\newfontfamily\greekfont{GFS Artemisia}[Ligatures=TeX,Script=Greek]

\begin{document}
This is English with some Greek: \textgreek{γέγονεν}.
Isn't that quite nice? εὕρηκα!
\end{document}

The GFS fonts all provide Greek since they are from the Greek Font Society. However, the above example does not work correctly in XeLaTeX.

In contrast, GFS Bodoni works with either XeLaTeX or LuaLaTeX:

\documentclass[11pt,a4paper]{scrartcl}
\usepackage{polyglossia}
\setmainlanguage[variant=british]{english}
\setotherlanguage{greek}
\usepackage{fontspec}
\setmainfont{GFS Bodoni}[Ligatures=TeX]
\newfontfamily\greekfont{GFS Bodoni}[Ligatures=TeX,Script=Greek]

\begin{document}
This is English with some Greek: \textgreek{γέγονεν}.
Isn't that quite nice? εὕρηκα!
\end{document}

You could, of course, still use TeX Gyre Termes as the main font for Latin scripts, but then you need to think about selecting a complementary font for Greek. So if you can find a font you like which supports both Latin and Greek scripts, life is probably easier.

I thought that XITS would make a good alternative to Termes since they are both Times clones. Unfortunately, though, XITS does not give the correct output for either XeLaTeX or LuaLaTeX, so it seems not a good choice, even though it claims to support Greek.

TeX Gyre Termes has very limited support for Greek. You can try the Tempora font, just included in TeX Live with today's update.

\documentclass[11pt,a4paper]{scrartcl}

\usepackage{fontspec}
\setmainfont[Ligatures=TeX]{Tempora}

\begin{document}

This is English with some Greek: γέγονεν.

Isn't that quite nice? εὕρηκα!

\end{document}

For single words, you can just type them; for more complex phrases or complete sentences, marking up the text with polyglossia commands or environments is highly recommended.

## Update

As of today's (2015-10-01) update, Tempora seems to behave also with LuaLaTeX

\documentclass[11pt,a4paper]{scrartcl}

\usepackage{fontspec}
\usepackage{polyglossia}

\setmainlanguage{english}
\setotherlanguage[variant=ancient]{greek}

\setmainfont{Tempora}
%\newfontfamily\greekfont[Script=Greek]{Tempora}

\begin{document}
This is English with some Greek: \textgreek{γέγονεν}.

Also without \verb|\textgreek|: γέγονεν

Isn't that quite nice? \textgreek{εὕρηκα}!

\end{document}

• The example works as desired with XeLaTeX but with LuaLaTeX it produces This is English with some Greek: γγονεν. Isn’t that quite nice? εὕρηκα! – Joe Corneli Sep 19 '15 at 23:28
• @JoeCorneli I checked and both XeLaTeX and LuaLaTeX load the same font file. However, if I look at the font in “Font Book” and there is something quite strange. What seems to happen is that “epsilon with tonos” is rendered with “epsilon with oxia” by XeLaTeX, but not by LuaLaTeX. More strangely, “epsilon with tonos” exists only in Tempora Italic. – egreg Sep 20 '15 at 7:51
• @JoeCorneli I saw in the TeX Live repository the announcement for a new version of the Tempora fonts: tug.org/svn/texlive/trunk/Master/texmf-dist/doc/fonts/tempora/… It should be available by tomorrow (2015-10-01) on all mirrors. – egreg Sep 30 '15 at 22:00
• @JoeCorneli I added the test for LuaLaTeX with the result after updating Tempora. The bug seems to have been fixed. – egreg Oct 1 '15 at 7:36

The font TeX Gyre Termes doesn't have the Greek character U+03AD"GREEK SMALL LETTER EPSILON WITH TONOS".

If you want to, you can fake it by adding U+0301 "COMBINING ACUTE ACCENT":

\documentclass{article}
\usepackage{fontspec}
\setmainfont{TeX Gyre Termes}
\begin{document}
This is English with some Greek: γε\char"0301γονεν.
\end{document}

Produces:

The accent doesn't look quite right, but that's because the font also doesn't have U+0384 "GREEK TONOS" which you would normally use for Greek. As you can understand, you simply shouldn't use TeX Gyre Termes for Greek. You should pick another font.

• OK. It's a pity, because the the Termes Greek letter forms are really beautiful. Thanks for the info! – Joe Corneli Sep 20 '15 at 13:42


text:
# People eating and gaining energy

Let's say we have 100 grams of chocolate which has 571 kilocalories (so it's about 2.4 megajoules) and we have barbell which weights 100 kilograms and need to lift it to 0.5 meters height.

So how many times you can lift it eating only 100 grams of chocolate? Let's use potential formula $$E=mgh$$ n – how many times you lift $$E=nmgh$$ $$n=\frac{E}{mgh}$$ $$n=\frac{2.4\cdot 10^6J}{100kg\cdot10\frac{m}{s^2}\cdot0.5m}=4800$$ So I get 4800 times, but I doubt it's true. So where and how many (approximately) our gained energy disappears, I mean becomes other kind of energy?

-
You are ignoring the energy exchanges of the body.The cells in the muscles have to eat energy and keep the temperature at 37C and there is infrared radiation in addition to chemical losses and lifting themselves up along with the barbell. This question is for a biology/physiology forum. –  anna v Jul 12 '11 at 15:38