text:
Q: How to concatenate matrices of unequal dimensions in 3rd dimension in MATLAB? Like the title says, I have several matrices that I would like to stack in the 3rd dimension. I currently have a solution for stacking unequal vectors horizontally, which reads like so:
A = [1;2;3;4];
B = [1;2];
[i1,j1] = ndgrid(1:size(A,1),1:size(A,2));
[i2,j2] = ndgrid(1:size(B,1),(1:size(B,2))+size(A,2));
plane_stats = accumarray([i1(:),j1(:);i2(:),j2(:)],[A(:);B(:)]);

for any two vectors A and B of unequal size (caveat: I have only tested this for column vectors, stacked horizontally). Wherever there would be a gap, this adds zeros as padding, so the result looks as follows:
1 1
2 2
3 0
4 0

However, I would like to generalize this to 3D so I can stack a third (also unequal) matrix C behind the combination of A and B, like so:
C = [1 2;3 4;5 6]
%some modification of existing ndgrid code
combo(:,:,1) =
1 1
2 2
3 0
4 0

combo(:,:,2) =
1 2
3 4
5 6
0 0

However I am not sure how to extend the solution I currently have. Any and all help/insight is appreciated

A: I would suggest something like this:
Given unequal sized 2D matrices A, B, C,
A = [1;2;3];
B = [1 2 3];
C = [1 2;3 4;5 6;7 8];

Solution 1
To stack at the 3rd dimension, we can simply do this:
n_rows = max([size(A,1),size(B,1),size(C,1)]);
n_cols = max([size(A,2),size(B,2),size(C,2)]);
combo = zeros(n_rows, n_cols, 3);
combo(1:size(A,1),1:size(A,2),1) = A
combo(1:size(B,1),1:size(B,2),2) = B
combo(1:size(C,1),1:size(C,2),3) = C

Solution 2
In additionally, to stack matrices incrementally in a loop:
matrices = {A,B,C};
for i = 1:numel(matrices)
    mat = matrices{i};
    combo(1:size(mat,1),1:size(mat,2),i) = mat;
end

Matlab will automatically padding zeros for you.
Solution 3
Another version without warning:
matrices = {A,B,C};
n_rows = 0; n_cols = 0;
for i = 1:numel(matrices)
    mat = matrices{i};
    n_rows = max(n_rows, size(mat,1));
    n_cols = max(n_cols, size(mat,2));
end
combo = zeros(n_rows, n_cols, numel(matrices));
for i = 1:numel(matrices)
    combo(1:size(matrices{i},1),1:size(matrices{i},2),i) = matrices{i};
end

Result
All above 3 solutions resulting the same:
>> combo

combo(:,:,1) =

     1     0     0
     2     0     0
     3     0     0
     0     0     0

combo(:,:,2) =

     1     2     3
     0     0     0
     0     0     0
     0     0     0

combo(:,:,3) =

     1     2     0
     3     4     0
     5     6     0
     7     8     0

A: Your approach for merging A and B is totally fine, it doesn't require any improvement and you can keep on using it:
A = [1;2;3;4];
B = [1;2];
[i1,j1] = ndgrid(1:size(A,1),1:size(A,2));
[i2,j2] = ndgrid(1:size(B,1),(1:size(B,2))+size(A,2));
plane_stats = accumarray([i1(:),j1(:);i2(:),j2(:)],[A(:);B(:)]);

Now... what's great about Matlab is that matrices are automatically expanded (and zero-padded) whenever the code requires it. Hence, for a given matrix C, all you have to do is:
combo = plane_stats;

[C_m,C_n] = size(C);

combo(1:C_m,1:C_n,2) = C;

A few examples:
combo = plane_stats;

C = [1 2; 3 4; 5 6];
[C_m,C_n] = size(C);

combo(1:C_m,1:C_n,2) = C

combo(:,:,1) =
     1     1
     2     2
     3     0
     4     0
combo(:,:,2) =
     1     2
     3     4
     5     6
     0     0

combo = plane_stats;

C = [1 2; 3 4; 5 6; 7 8; 9 10];
[C_m,C_n] = size(C);

combo(1:C_m,1:C_n,2) = C

combo(:,:,1) =
     1     1
     2     2
     3     0
     4     0
     0     0
combo(:,:,2) =
     1     2
     3     4
     5     6
     7     8
     9    10

combo = plane_stats;

C = [1 2 3; 3 4 5; 5 6 7; 7 8 9; 10 11 12; 13 14 15];
[C_m,C_n] = size(C);

combo(1:C_m,1:C_n,2) = C

combo(:,:,1) =
     1     1     0
     2     2     0
     3     0     0
     4     0     0
     0     0     0
     0     0     0
combo(:,:,2) =
     1     2     3
     3     4     5
     5     6     7
     7     8     9
    10    11    12
    13    14    15

combo = plane_stats;

C = [1; 2];
[C_m,C_n] = size(C);

combo(1:C_m,1:C_n,2) = C

combo(:,:,1) =
     1     1
     2     2
     3     0
     4     0
combo(:,:,2) =
     1     0
     2     0
     0     0
     0     0


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Turnitin Acquires ExamSoft from Spectrum Equity
By Rebecca Hinkle
Turnitin announced on Oct 21st, that they had acquired ExamSoft from Spectrum Equity. ExamSoft, a software platform that provides exam functionality and security for educational institutions, will be working with Turnitin to secure and monitor educational institutions for flexibility for assessments at educational and professional levels.
"Turnitin and ExamSoft share a passion for student success and delivering innovative assessments," said Chris Caren, the CEO of Turnitin, in a statement.
With ExamSoft and Turnitin working together, educational institutions will be able to ensure and promote academic integrity while also allowing the use of a combination of services to help students along the way.
How Turnitin Will Leverage ExamSoft's Platform
Turnitin was started by four university students in 1998, and was launched two years later as a plagiarism prevention service. In 2002, Turnitin debuted an online grading platform for educators. Since then, the company has had over 26 million students and educators use their site.
ExamSoft is an education software development program from Dallas, Texas that creates software solutions that focus on providing high levels of security for educational and certification institutions. With their assessment platform, ExamSoft can enable the educators to create, administer, grade, and analyze assignments. Data reporting shows course and student performing results that can show educators what needs to change in how their students are learning, tracking student engagement, and find curriculum areas that need improvement.
Welcome to the Turnitin family, @ExamSoft! https://t.co/HJMCo2gAvr pic.twitter.com/5uKGWdBm2e
— Turnitin (@Turnitin) October 22, 2020
"We look forward to working with the Turnitin team to continue to advance integrity solutions across all levels and modalities of education to improve learning outcomes," said ExamSoft CEO Sebastian Vos, in a statement. ExamSoft has a 98% client retention rate, over 1,900 worldwide programs, and has delivered more than 63 million exams.
Spectrum Equity is a company that is a leading growth equity firm that has been providing capital and strategic support to innovative Internet, software, and information services companies for over 25 years.
Academic Dishonesty with Distance Learning
In this time of distance learning, it is crucial to identify and stop students committing plagiarism on written papers and other assignments. It can be easily avoided if the students have an understanding about plagiarism and paraphrasing.
One study "Plagiarism Instruction Online: Assessing Undergraduate Students' Ability to Avoid Plagiarism" by Pamela A. Jackson discusses students' understanding of plagiarism in a traditional educational setting. The study shared the results of how students paraphrased certain sections from the test.
"Most students did provide a citation for their paraphrase; however, in general, the paraphrases followed the original passage too closely. Students often used exact language from the original passage without using quotation marks, rearranged the sentence structure of the original passage, or simply replaced certain words to make it their own," reported Jackson in the study.
Students having issues with plagiarism was concern in the classroom before COVID-19 changed schools to remote learning. Now with remote learning, students could have more difficulties understanding when they are committing plagiarism in their work. With Turnitin and ExamSoft, more educational institutions will be able to promote and encourage academic integrity, original work, and independent research from students as they continue to study remotely.
Correction October 23: A previous version of this article misspelled the name of the previous owners of ExamSoft. They are Spectrum Equity, not Spectrum Effect.
Featured Image: Scott Robison, Flickr.
Could Algorithms Replace Teachers?
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Is Remote Learning the Future of Education?
Is Remote Learning the Future of Education? - e-Learning Feeds says:
[…] Read the full story by eLearningInside News […]

redpajama_set_name: RedPajamaCommonCrawl


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text:
package cache

import (
	"context"
	"go/token"
	"path/filepath"
	"strings"
	"sync"

	"golang.org/x/tools/internal/lsp/source"
	"golang.org/x/tools/internal/span"
)

// viewFile extends source.File with helper methods for the view package.
type viewFile interface {
	source.File

	filename() string
	addURI(uri span.URI) int
}

// fileBase holds the common functionality for all files.
// It is intended to be embedded in the file implementations
type fileBase struct {
	uris  []span.URI
	fname string
	kind  source.FileKind

	view *view

	handleMu sync.Mutex
	handle   source.FileHandle
}

func basename(filename string) string {
	return strings.ToLower(filepath.Base(filename))
}

func (f *fileBase) URI() span.URI {
	return f.uris[0]
}

func (f *fileBase) filename() string {
	return f.fname
}

// View returns the view associated with the file.
func (f *fileBase) View() source.View {
	return f.view
}

// Content returns a handle for the contents of the file.
func (f *fileBase) Handle(ctx context.Context) source.FileHandle {
	f.handleMu.Lock()
	defer f.handleMu.Unlock()
	if f.handle == nil {
		f.handle = f.view.Session().GetFile(f.URI())
	}
	return f.handle
}

func (f *fileBase) FileSet() *token.FileSet {
	return f.view.Session().Cache().FileSet()
}

redpajama_set_name: RedPajamaGithub


Medical Marijuana, Inc.'s Q3 2018 revenue exceeded $16.8 million dollars, an increase of more than 116% over Q3 2017. Gross profit for Q3 2018 increased nearly 140% over Q3 2017. This announcement continues on a trend of successful growth for Medical Marijuana, Inc. and its portfolio of companies, including Kannaway.
Gross profit for the company for the nine-months ending September 30, 2018 increased nearly 140% over the prior year nine-month period. Revenue for the nine-month period ending September 30, 2018 exceeded $42.1 million dollars, an increase of 146% over the prior year nine-month period.
Kannaway contributed to Medical Marijuana, Inc.'s success in Q3 through a number of initiatives that added to our own trend of upward growth.
We also added new athletes to the Kannaway Sports team, including Olympic Gold Medalist Amy Van Dyken, MLB star pitcher Goose Gossage, and Super Bowl champion Derek Kennard.
Kannaway launched new products in Q3, like Dante's Pure Hemp Oil, and Hempy's Leggings.
In preparation of our Europe Grand Opening Event in November, Kannaway hosted a series of regional events as part of our tour across Europe, hitting cities like Copenhagen, Oslo, Bremen, Milan, Rome, and more.
Our hard work didn't go unnoticed. Kannaway was featured in media outlets that include The Street, Benzinga, Civilized, the Wall Street Journal, and Direct Selling News.
You can read the full press release here or visit our news feed for the latest in Kannaway news.
Kannaway Hemp for Humanity Sale EXTENDED for 24 Hours!


text:
package aws

import (
	"fmt"
	"log"
	"time"

	"github.com/aws/aws-sdk-go/aws"
	"github.com/aws/aws-sdk-go/aws/awserr"
	"github.com/aws/aws-sdk-go/service/elasticbeanstalk"
	"github.com/hashicorp/terraform/helper/schema"
)

func resourceAwsElasticBeanstalkApplicationVersion() *schema.Resource {
	return &schema.Resource{
		Create: resourceAwsElasticBeanstalkApplicationVersionCreate,
		Read:   resourceAwsElasticBeanstalkApplicationVersionRead,
		Update: resourceAwsElasticBeanstalkApplicationVersionUpdate,
		Delete: resourceAwsElasticBeanstalkApplicationVersionDelete,

		Schema: map[string]*schema.Schema{
			"application": &schema.Schema{
				Type:     schema.TypeString,
				Required: true,
				ForceNew: true,
			},
			"description": &schema.Schema{
				Type:     schema.TypeString,
				Optional: true,
			},
			"bucket": &schema.Schema{
				Type:     schema.TypeString,
				Required: true,
				ForceNew: true,
			},
			"key": &schema.Schema{
				Type:     schema.TypeString,
				Required: true,
				ForceNew: true,
			},
			"name": &schema.Schema{
				Type:     schema.TypeString,
				Required: true,
				ForceNew: true,
			},
			"force_delete": &schema.Schema{
				Type:     schema.TypeBool,
				Optional: true,
				Default:  false,
			},
		},
	}
}

func resourceAwsElasticBeanstalkApplicationVersionCreate(d *schema.ResourceData, meta interface{}) error {
	conn := meta.(*AWSClient).elasticbeanstalkconn

	application := d.Get("application").(string)
	description := d.Get("description").(string)
	bucket := d.Get("bucket").(string)
	key := d.Get("key").(string)
	name := d.Get("name").(string)

	s3Location := elasticbeanstalk.S3Location{
		S3Bucket: aws.String(bucket),
		S3Key:    aws.String(key),
	}

	createOpts := elasticbeanstalk.CreateApplicationVersionInput{
		ApplicationName: aws.String(application),
		Description:     aws.String(description),
		SourceBundle:    &s3Location,
		VersionLabel:    aws.String(name),
	}

	log.Printf("[DEBUG] Elastic Beanstalk Application Version create opts: %s", createOpts)
	_, err := conn.CreateApplicationVersion(&createOpts)
	if err != nil {
		return err
	}

	d.SetId(name)
	log.Printf("[INFO] Elastic Beanstalk Application Version Label: %s", name)

	return resourceAwsElasticBeanstalkApplicationVersionRead(d, meta)
}

func resourceAwsElasticBeanstalkApplicationVersionRead(d *schema.ResourceData, meta interface{}) error {
	conn := meta.(*AWSClient).elasticbeanstalkconn

	resp, err := conn.DescribeApplicationVersions(&elasticbeanstalk.DescribeApplicationVersionsInput{
		ApplicationName: aws.String(d.Get("application").(string)),
		VersionLabels:   []*string{aws.String(d.Id())},
	})
	if err != nil {
		return err
	}

	if len(resp.ApplicationVersions) == 0 {
		log.Printf("[DEBUG] Elastic Beanstalk application version read: application version not found")

		d.SetId("")

		return nil
	} else if len(resp.ApplicationVersions) != 1 {
		return fmt.Errorf("Error reading application version properties: found %d versions of label %q, expected 1",
			len(resp.ApplicationVersions), d.Id())
	}

	if err := d.Set("description", resp.ApplicationVersions[0].Description); err != nil {
		return err
	}

	return nil
}

func resourceAwsElasticBeanstalkApplicationVersionUpdate(d *schema.ResourceData, meta interface{}) error {
	conn := meta.(*AWSClient).elasticbeanstalkconn

	if d.HasChange("description") {
		if err := resourceAwsElasticBeanstalkApplicationVersionDescriptionUpdate(conn, d); err != nil {
			return err
		}
	}

	return resourceAwsElasticBeanstalkApplicationVersionRead(d, meta)

}

func resourceAwsElasticBeanstalkApplicationVersionDelete(d *schema.ResourceData, meta interface{}) error {
	conn := meta.(*AWSClient).elasticbeanstalkconn

	application := d.Get("application").(string)
	name := d.Id()

	if d.Get("force_delete").(bool) == false {
		environments, err := versionUsedBy(application, name, conn)
		if err != nil {
			return err
		}

		if len(environments) > 1 {
			return fmt.Errorf("Unable to delete Application Version, it is currently in use by the following environments: %s.", environments)
		}
	}
	_, err := conn.DeleteApplicationVersion(&elasticbeanstalk.DeleteApplicationVersionInput{
		ApplicationName:    aws.String(application),
		VersionLabel:       aws.String(name),
		DeleteSourceBundle: aws.Bool(false),
	})

	if err != nil {
		if awserr, ok := err.(awserr.Error); ok {
			// application version is pending delete, or no longer exists.
			if awserr.Code() == "InvalidParameterValue" {
				d.SetId("")
				return nil
			}
		}
		return err
	}

	d.SetId("")
	return nil
}

func resourceAwsElasticBeanstalkApplicationVersionDescriptionUpdate(conn *elasticbeanstalk.ElasticBeanstalk, d *schema.ResourceData) error {
	application := d.Get("application").(string)
	description := d.Get("description").(string)
	name := d.Get("name").(string)

	log.Printf("[DEBUG] Elastic Beanstalk application version: %s, update description: %s", name, description)

	_, err := conn.UpdateApplicationVersion(&elasticbeanstalk.UpdateApplicationVersionInput{
		ApplicationName: aws.String(application),
		Description:     aws.String(description),
		VersionLabel:    aws.String(name),
	})

	return err
}

func versionUsedBy(applicationName, versionLabel string, conn *elasticbeanstalk.ElasticBeanstalk) ([]string, error) {
	now := time.Now()
	resp, err := conn.DescribeEnvironments(&elasticbeanstalk.DescribeEnvironmentsInput{
		ApplicationName:       aws.String(applicationName),
		VersionLabel:          aws.String(versionLabel),
		IncludeDeleted:        aws.Bool(true),
		IncludedDeletedBackTo: aws.Time(now.Add(-1 * time.Minute)),
	})

	if err != nil {
		return nil, err
	}

	var environmentIDs []string
	for _, environment := range resp.Environments {
		environmentIDs = append(environmentIDs, *environment.EnvironmentId)
	}

	return environmentIDs, nil
}


The mover of the debate, MP Fiona Bruce voiced her profound alarm at the increasing level of persecution of Christians in a number of Muslim countries across the world mostly due to the whipping up of religious hatred by politically motivated groups.
But it was her colleague, MP Tom Greatrex who drew attention to Malaysia.
Greatrex, who is a member of the All-Party Committee on Malaysia, made special referrence to the predicament of the Christians in East Malaysia.
"The report to which she (Bruce) refers does not cover numerous countries, including Malaysia, which is not in the Middle East.
"The situation there (Malaysia) is of significant concern. The recent decision in Malaysia to ban Christians from using the word "Allah" which has been used in Malay as a term for God for centuries.
"It (Malaysia) has effectively outlawed the Bible, particularly in the Christian eastern states of Malaysia.
"Is she (Bruce) concerned about the wider ramifications in other parts of the world not covered by the report that she cites?" Greatrex asked.
Online portal Sarawak Report which noted the debate said UK parliament examined how appalling situations can happen when countries with citizens of different religions and races permit and support extremism and prejudice, a path that certain factions are trying to initiate in Malaysia.
The report noted that accusations against non-Muslims and liberal Muslims have started to be plentiful and practices accepted for centuries are currently being damned and outlawed.
The Herald to use the term Allah in its Malay publications.
The decision has brought to boil simmering discontent at the suppressing policies of the Umno-led government and the blatant disregard for individual's constitutional right to freely practice their own religion.
Does it mean, someone was telling lies in England not so long ago?


text:
\section{Introduction}
\setcounter{equation}{0}

Recently, we know two major sources of detecting non-integrability
of Hamiltonian systems. One is Poincar\'e's theorem\cite{4poin}
 of nineteenth century and the other is
Ziglin's analysis\cite{zig} relatively recently found in the connection with
 the classical singularity analysis\cite{rgb} by
Fuchs, Kowalevskaya and Painlev\'e. What matters here is the following thing.
The two different methods have
significantly different  predictive capacities to tell non-integrability.
While Poincar\'e's theorem is asserting the non-existence of additional
conserved quantities such as
\begin{equation}
\label{eq:pphi}
  \Phi_{0}(\{q_{\nu}\})+\mu \Phi_{1}(\{q_{\nu},p_{\nu}\}) +
  \mu^{2} \Phi_{2}(\{q_{\nu},p_{\nu}\})+\cdots,
\end{equation}
 for a Hamiltonian system perturbed from an integrable systems
\begin{equation}
  H=H_{0}(\{q_{\nu}\})+\mu H_{1}(\{q_{\nu},p_{\nu}\}) +
  \mu^{2} H_{2}(\{q_{\nu},p_{\nu}\})+\cdots,
\end{equation}
 where \(\mu\) is a coupling constant whose explicit value is unknown,
 Ziglin's type theorem can prove the non-existence of additional
analytic integral for an explicit Hamiltonian.
To illustrate the differences between them  more concretely,
we can give the following examples.
The restricted three-body problem was proved to be non-integrable
by Whittaker in 1904\cite{3whittaker} based on Poincar\'e's theorem,
while Ziglin's theorem have never proved its non-integrability
because of the problem that a
Hamiltonian of the restricted three-body problem cannot be  defined
without the infinitesimal operation \((m\rightarrow 0)\) and we don't
have any
explicit Hamiltonian which is necessary to perform  solid Ziglin's analysis.
On the other hand, Poincar\'e's theorem assumes the condition on the
Hessian of Hamiltonian, i.e.,
the Hessian of Hamiltonian
should not vanish
\begin{equation}
   |\frac{\partial^{2} H_{0}}{\partial q_{\nu} \partial q_{\lambda}}|
   \neq 0.
\end{equation}

By this assumption, Poincar\'e's theorem cannot not tell
anything about the non-integrability of nonlinear lattices consisting of
harmonic  terms \(H_{0}=\frac{1}{2}\sum_{i=1}^{n}q^{2}_{i}\)
plus unharmonic terms such as the nonlinear
lattices of Fermi-Pasta-Ulam type (FPU lattices). Therefore, in the present
paper, we examine the possibility of non-integrability proof of
FPU lattices via Ziglin's analysis.
It should be also mentioned that Ziglin's analysis
 have some technical difficulties, as is illustrated later, and
no  non-integrability proof has been so far
studied either for realistic Hamiltonian systems
such as  the FPU lattices with non-homogeneous
potential functions,
except few rare instances
such as the Henon-Heiles systems with two degrees of
freedom\cite{henon,zig1,ito}
or for Hamiltonian systems with many degrees of freedom besides the case of
 Hamiltonian systems with global
and symmetric coupling\cite{ku1}.

The purpose of the present paper is to show that
 this  Ziglin's analysis
can be performed for checking the non-integrability of
 nonlinear lattices including the FPU lattices and to show that
these nonlinear  lattices have no other
analytic conserved
quantities.

In Section 2,
we show that singularity analysis can be
done for a class of FPU lattices by
giving the explicit eigenvalues of the monodromy matrices about a
special solution in terms of  elliptic functions.

In Section 3,
we prove a
non-integrability theorem for  FPU lattices
based on the singularity analysis in Section 2,
which shows  that
each FPU lattice has no additional analytic conserved quantities in the
low energy limit.
In Section 4 we discuss the classification of
non-homogeneous nonlinear lattices via the degree of non-integrability.

In Section 5, we give a brief summary and  open questions about the present
analysis.

\section{Singularity analysis of nonlinear lattices}
\setcounter{equation}{0}

 Consider the following one-dimensional lattice:
\begin{equation}
\label{eq:general}
  H=\frac{1}{2}\sum_{i=1}^{n}p_{i}^{2}+
  \sum_{i=1}^{n+1}\upsilon(q_{i-1}-q_{i}).
\end{equation}
 If  this Hamiltonian admits the reflective invariance, namely, if it is
 invariant under the involutive symplectic diffeomorphism,
\begin{equation}
  J_{r}H=H,
\end{equation}
 where
\begin{equation}
 J_{r}: \mbox{\boldmath$C$}^{2n}\rightarrow \mbox{\boldmath$C$}^{2n},
 J_{r}:(q_{1},\cdots,q_{n},p_{1},\cdots
 ,p_{n})\mapsto (-q_{1},\cdots,-q_{n},-p_{1},\cdots,-p_{n}),
\end{equation}
 the potential function \(\upsilon(X)\) must satisfy the condition given by
\begin{equation}
  \upsilon(X)=\upsilon(-X).
\end{equation}
 In the case of harmonic lattices, we have
 \(\upsilon(X)=\frac{1}{2}\mu_{2}X^{2}\).
 Thus more
 general nonlinear lattices with the reflection symmetry can be defined
 by the
 following
 polynomial function of even degree in \(X\):
\begin{equation}
\label{eq:potential}
  \upsilon(X)=\frac{\mu_{2}}{2}X^{2}+
  \frac{\mu_{4}}{4}X^{4}+\cdots +\frac{\mu_{2m}}{2m}
  X^{2m}.
\end{equation}
In this paper, however,  we assume that this potential function is a polynomial
  function of degree  \(4\), namely, we consider the
 Fermi-Pasta-Ulam(FPU) lattices\cite{fpu}
\begin{equation}
\label{eq:ulam}
  H_{FPU}=\frac{1}{2}\sum_{i=1}^{n}p_{i}^{2}+\frac{\mu_{2}}{2}
  \sum_{i=1}^{n+1}(q_{i-1}-q_{i})^{2}+
  \frac{\mu_{4}}{4}\sum_{i=1}^{n+1}(q_{i-1}-q_{i})^{4}.
\end{equation}

Imposing the fixed boundary condition as
\begin{equation}
\label{eq:fixbound}
  q_{0}=q_{n+1}=0, \quad n=\mbox{odd},
\end{equation}
the equations of motion are given by
\begin{equation}
\label{eq:nls}
\begin{array}{l}
  \ddot{q_{1}}=  -\mu_{2}(2q_{1}-q_{2})-\mu_{4}(q_{1}^{3}+(q_{1}-q_{2})^{3}),\\
  \ddot{q_{2}}=  -\mu_{2}(2q_{2}-q_{1}-q_{3})-\mu_{4}((q_{2}-q_{1})^{3}+
   (q_{2}-q_{3})^{3}),\\
  \vdots  \cdots\\
  \vdots  \cdots\\
  \ddot{q_{n}}=
  -\mu_{2}(2q_{n}-q_{n-1})-\mu_{4} ((q_{n}-q_{n-1})^{3}+q_{n}^{3})
\end{array}
\end{equation}

Define \(M'=\{(\mbox{\boldmath$q,p$})\in
\mbox{\boldmath$C$}^{2n}|q_{1}\ne 0 \mbox{ or } p_{1}\ne 0\}\),
\(\hat{M}=M'/J\) and
the canonical projection \(\pi:M'\rightarrow \hat{M}\).
An additional complex analytic integral in \(\mbox{\boldmath$C$}^{2n}\),
 if it exists, also induces the corresponding analytic integral
 on \(\hat{M}\). Therefore it is sufficient to check the
 non-integrability of  the reduced equations of
 FPU lattices on \(\hat{M}\) in order to confirm the
 non-integrability of the original equations of FPU lattices.

Consider the following one-parameter family of the solution for (\ref{eq:nls})
\begin{equation}
\label{eq:solsol1}
\Gamma(\epsilon,t):  q_{j}=\frac{((-1)^{j}-1)}{2}(-1)^{\frac{j+1}{2}}C\phi(t)
\end{equation}
 with the initial condition
\begin{equation}
\label{eq:initial}
    \phi (0)=1, \quad \dot{\phi}(0)=0,\quad H_{FPU}(0)=\epsilon.
\end{equation}
This is a complex solution in the complex time plane as
\begin{equation}
\label{eq:compsol}
 q_{1}=C\phi(t),q_{2}=0,q_{3}=-C\phi(t),\cdots,q_{n-1}=0,
q_{n}=(-1)^{\frac{n-1}{2}}
 C\phi(t),
\end{equation}
 which is depicted in Fig. 1.
  Thus, the underlying equation of \(\phi (t)\) is reduced to the
  following Hamiltonian
  system with one degree of freedom:
\begin{equation}
\label{eq:oneh}
  \ddot{\phi}+2\mu_{2}\phi+2\mu_{4} C^{2}\phi^{3}=0,
\end{equation}
whose Hamiltonian is given by
\begin{equation}
\label{eq:determin}
   H(\phi,\dot{\phi})=
  \frac{1}{2}(\dot{\phi})^{2}+\mu_{2}\phi^{2}+\frac{\mu_{4} C^{2}}{2}\phi^{4}
  =\mbox{Const.}
\end{equation}
 Then the original total energy \(\epsilon\) is written as follows:
\begin{equation}
\label{eq:energyr}
  \epsilon=H=
  H(\phi,\dot{\phi})\frac{n+1}{2}C^{2}=
  \frac{n+1}{2}C^{2}(\mu_{2}+\frac{1}{2}\mu_{4} C^{2})
\end{equation}
by  the initial condition (\ref{eq:initial}).
This equation (\ref{eq:energyr}) enable us to choose
 \(C\) as the following {\it complex} quantity:
\begin{equation}
\label{eq:cdterm}
  C=\sqrt{\frac{\sqrt{\mu^{2}_{2}+\frac{4\epsilon}{n+1}\mu_{4}}
  -\mu_{2}}
  {\mu_{4}}}.
\end{equation}
Therefore, \(H(\phi,\dot{\phi})\) becomes
\begin{equation}
  H(\hat{q},\hat{p})=\frac{1}{2}\hat{p}^{2}+\mu_{2}\hat{q}^{2}
  +\frac{\sqrt{\mu_{2}^{2}+\frac{4\epsilon}{n+1}\mu_{4}}-\mu_{2}}{2}\hat{q}^{4},
\end{equation}
 where \(\hat{q}=\phi,\hat{p}=\dot{\phi}\).
By combining (\ref{eq:determin}) with (\ref{eq:energyr}), we obtain
the
differential equation of \(\phi(t)\) as
\begin{equation}
\label{eq:finalde}
 \frac{1}{2}(\dot{\phi})^{2}=\gamma_{2}
 (1-\phi^{2})+\frac{\gamma_{4}}{2}(1-\phi^{4})
\end{equation}
with the  system parameters
\begin{equation}
  \gamma_{2}\equiv \mu_{2},\quad \gamma_{4}\equiv \mu_{4}C^{2}.
\end{equation}
Thus, the phase curve \(\Gamma(t,\epsilon)\)(\ref{eq:solsol1}) is given by
the {\it elliptic} integral
\begin{equation}
\int_{\phi(0)=1}^{\phi(t)}
\frac{d\phi}{\sqrt{2}\sqrt{\gamma_{2}
(1-\phi^{2})+\frac{\gamma_{4}}{2}(1-\phi^{4})
}}=t
\end{equation}
 and
the solution of
this differential equation (\ref{eq:finalde}) with the condition
\begin{equation}
 \gamma_{2m=4}\ne 0
\end{equation}
 is given explicitly by the formula
\begin{equation}
  \phi(t)=cn(k;\alpha t),
\end{equation}
 where
\begin{equation}
\label{eq:3rel1}
   \alpha=\sqrt{2\gamma_{2}+2\gamma_{4}},\quad
   k=\sqrt{\frac{\gamma_{4}}{2\gamma_{2}+2\gamma_{4}}},
\end{equation}
\(cn(k;\alpha t)\) is the  Jacobi
{\it cn} elliptic
function, and \(k\) is  the modulus of the elliptic integral, because
of
the following equalities
\begin{equation}
\begin{array}{l}
(\frac{d\phi}{dt})^{2}=\{-\alpha sn(k;\alpha t)dn(k;\alpha t)\}^{2}\\
		      =\alpha^{2}\{(-2k^{2}+1)(1-cn^{2}(k;\alpha t))
		      +k^{2}(1-cn^{4}(k;\alpha t))\}\\
		      =2\gamma_{2}(1-\phi^{2})+\gamma_{4}(1-\phi^{4})
\end{array}
\end{equation}
in Eq. (\ref{eq:finalde}).
We remark here that
the relation
\begin{equation}
 \gamma_{2}+\gamma_{4}=\mu_{2}+C^{2}\mu_{4}=
 \sqrt{\mu_{2}^{2}+\frac{4\epsilon}{n+1}\mu_{4}}>0,
\end{equation}
holds
for \(\mu_{4}>0,\mu_{2}> 0,\epsilon>0\). This means that
the modulus of the elliptic function \(k\) satisfies the following relation:
\begin{equation}
\label{eq:modulus}
  0< k=\frac{1}{\sqrt{2}}
\sqrt{1-\frac{1}{\sqrt{1+\frac{4\epsilon\mu_{4}}{(n+1)\mu^{2}_{2}}}}}
<\frac{1}{\sqrt{2}}
\end{equation}
 and \(C\) is a real quantity.
On the contrary, if the following relations
\begin{equation}
 \mu_{2}>0, \mu_{4}>0, \epsilon<0
\end{equation}
are satisfied,
the modulus of the elliptic function \(k\) becomes pure imaginary, i.e.,
  \(k^{2}<0\).  \(C\) becomes also pure imaginary as in Fig.1. We note here
that even in this {\it unphysical} case, there is no barrier to perform
the following singularity analysis. However, in the following we shall
consider the physical case with the condition as
\(\mu_{2}>0,\mu_{4}>0, \epsilon>0\).
According to the reduction \(\pi : M'\rightarrow \hat{M}=M'/J\) by the
reflective symmetry \(J\),
 the solution (\ref{eq:compsol})
of the induced FPU lattices on \(\hat{M}\) for \(\mu_{4}>0,\mu_{2}\geq 0\)
is a single-valued, meromorphic and doubly periodic function in \(t\),
having the two {\it fundamental periods} in the complex
time plane as follows:
\begin{equation}
\label{eq:rel2}
    T_{1}(\epsilon,\mu)=\frac{2K(k)}{\alpha},
    \quad T_{2}(\epsilon,\mu)=\frac{2K(k)+2iK'(k)}{\alpha}.
\end{equation}
 Here \(K(k)\) and \(K'(k)\) are
 the complete elliptic integrals of the first kind:
\begin{equation}
   K(k)=\int_{0}^{1}\frac{dv}{\sqrt{(1-v^{2})(1-k^{2}v^{2})}},\quad
   K'(k)=\int_{0}^{1}\frac{dv}{\sqrt{(1-v^{2})(1-(1-k^{2})v^{2})}}.
\end{equation}
 Note here that these solutions have a {\it single} pole at
 \(t=\tau,\) where
 \(\tau=\frac{2K(k)}{\alpha}+i\frac{K'(k)}{\alpha}\quad (\mbox{mod }
 T_{1},T_{2})\)
 in the parallelogram of each period cell; the phase curves
 \(\Gamma(\epsilon,t)\) are punctured tori.
If we consider the solution (\ref{eq:compsol}) not on the
reduced phase space \(\hat{M}=M'/J\) but on the original
phase space \(M'\), the solution  has the following fundamental periods as
\begin{equation}
 T'_{1}(\epsilon,\mu)=\frac{4K(k)}{\alpha},
    \quad T'_{2}(\epsilon,\mu)=\frac{2K(k)+2iK'(k)}{\alpha},
\end{equation}
 which are different from (\ref{eq:rel2}) and this solution on
\(M'\) has
{\it two}  poles at \(t=\tau',\)\\ where
 \(\tau'=\frac{2K(k)}{\alpha}+i\frac{K'(k)}{\alpha},
\frac{4K(k)}{\alpha}+i\frac{K'(k)}{\alpha}
\quad (\mbox{mod }
 T'_{1},T'_{2})\)
in the parallelogram of each periodic cell.

 Let
 us consider the variational equations along the phase curve
 \(\Gamma(\epsilon,t)\) in (\ref{eq:solsol1}) of the general
 nonlinear lattices with the reflection symmetry.
 The linearized variational equations are written as follows:
\begin{equation}
\label{eq:firstve}
\begin{array}{l}
  \dot{\eta_{j}}=
  \ddot{\xi_{j}}=
  -\sum_{k=1}^{n}\left.\frac{\partial^{2}V}{\partial q_{k}\partial q_{j}}
  \right|_{\Gamma}
  \xi_{k}\\
  =
 -(\gamma_{2}+3\gamma_{4}cn^{2}(k;\alpha t))
 (2\xi_{j}-\xi_{j-1}-\xi_{j+1})
  \mbox{  for  }1\leq j\leq n,
\end{array}
\end{equation}
 where
   \(\xi_{0}=\xi_{n+1}=\eta_{0}=\eta_{n+1}=0\) and \(\xi_{j}=\delta q_{j},
   \eta_{j}=\delta p_{j}\quad(1\leq j\leq n)\).\\
  If we rewrite
  these linear
  variational equations (\ref{eq:firstve}) in the  form of the vector
   variational equation as
\begin{equation}
\frac{d^{2}}{dt^{2}}
\mbox{\boldmath$\xi$}=-(\gamma_{2}+3\gamma_{4}cn^{2}(k;\alpha t))
\left[
\begin{array}{ccccc}
2 & -1 & 0 & \cdots & 0 \\
-1 & 2 & -1 & \cdots & 0 \\
0 & -1 & 2 & -1 & \cdots \\
\cdots & \cdots & \cdots & \cdots & \cdots \\
0 & \cdots & 0 & -1 & 2,
\end{array}
\right]
\mbox{\boldmath$\xi$}
\end{equation}
and use the fact that  the eigenvalues of the \(n\times n\) symmetric matrix
\begin{equation}
\mbox{\boldmath$G$}=
\left[
\begin{array}{ccccc}
2 & -1 & 0 & \cdots & 0 \\
-1 & 2 & -1 & \cdots & 0 \\
0 & -1 & 2 & -1 & \cdots \\
\cdots & \cdots & \cdots & \cdots & \cdots \\
0 & \cdots & 0 & -1 & 2
\end{array}
\right]
\end{equation}
are obtained as \(\{4\mbox{sin}^{2}(\frac{j\pi}{2(n+1)})|1\leq
  j\leq n\}\)
 by a normal orthogonal transformation
\(\mbox{\boldmath$G$}\rightarrow
\mbox{\boldmath$OGO^{-1}$}\),
 the variational equations (\ref{eq:firstve}) can be transformed
 into the decoupled form:
\begin{equation}
\label{eq:cve}
  \ddot{\xi'_{j}}(t)=-4\mbox{sin}^{2}(\frac{j\pi}{2(n+1)})
  (\gamma_{2}+3\gamma_{4}cn^{2}(k;\alpha t))
  \xi'_{j}(t)\quad(1\leq j \leq n),
\end{equation}
where \(\mbox{\boldmath$\xi'$}=\mbox{\boldmath$O\xi$}\).
Clearly, these equations are
regarded as  the vector form of {\it Hill's equations}\cite{3hill}
\begin{equation}
  \frac{d^{2}\mbox{\boldmath$\xi'$}}{dt^{2}}+\mbox{\boldmath$A$}(t)
  \mbox{\boldmath$\xi'$}=0,\quad
  \mbox{\boldmath$A$}(t+T)=\mbox{\boldmath$A$}(t),
\end{equation}
 where \(T=T_{1},T_{2}\).
For \(j=\frac{n+1}{2}\), the relation
\begin{equation}
\label{eq:mannaka}
\xi'_{\frac{n+1}{2}}=\sqrt{\frac{2}{n+1}}(\xi_{1}-\xi_{3}+\xi_{5}+\cdots+(-1)^{\frac{n-1}{2}}
\xi_{n})
\end{equation}
holds. The corresponding variational equation
\begin{equation}
\label{eq:tangenve}
  \ddot{\xi'}_{\frac{n+1}{2}}=-2(\gamma_{2}+3\gamma_{4}cn^{2}(k;\alpha t))
  \xi'_{\frac{n+1}{2}}(t)
\end{equation}
has a time-dependent integral
\(I(\mbox{\boldmath$\xi$},\dot{\mbox{\boldmath$\xi$}};t)
\equiv I(\mbox{\boldmath$\xi$},\mbox{\boldmath$\eta$};t)\)
derived  from the variational
operation  \(\mbox{\boldmath$D$}\) applied to the Hamiltonian
(\ref{eq:ulam}):
\begin{equation}
\begin{array}{l}
  I(\mbox{\boldmath$\xi$},\mbox{\boldmath$\eta$};t)=
   \mbox{\boldmath$D$}H\equiv
  (\mbox{\boldmath$\eta$}\cdot
  \frac{\partial}{\partial\mbox{\boldmath$p$}}+\mbox{\boldmath$\xi$}\cdot
  \frac{\partial}{\partial\mbox{\boldmath$q$}})H
  =\mbox{\boldmath$\eta$}\cdot\mbox{\boldmath$p$}+\mbox{\boldmath$\xi$}
  \cdot\mbox{\boldmath$V_{q}$}\\
  =C\dot{\phi}(\eta_{1}-\eta_{3}+\eta_{5}+
  \cdots+(-1)^{\frac{n-1}{2}}\eta_{n})\\
  +2(C\gamma_{2}cn(k;\alpha t)+
  C\gamma_{4}cn^{3}(k;\alpha t))
  (\xi_{1}-\xi_{3}+\xi_{5}+\cdots+(-1)^{\frac{n-1}{2}}
  \xi_{n}),
\end{array}
\end{equation}
because
\begin{equation}
\begin{array}{l}
\frac{1}{C}\frac{dI}{dt}=\dot{\phi}(\ddot{\xi_{1}}-\ddot{\xi_{3}}+
\cdots+(-1)^{\frac{n-1}{2}}
\ddot{\xi_{n}})\\
+2\dot{\phi}(\gamma_{2}+3\gamma_{4}cn^{2}(k;\alpha t))
(\xi_{1}-\xi_{3}+\cdots+(-1)^{\frac{n-1}{2}}
\xi_{n})
=0.
\end{array}
\end{equation}
Eq. (\ref{eq:tangenve}) is called the {\it  tangential variational
equation}.
On the other hands, a \((2n-2)\)-dimensional
{\it normal variational equation}(NVE) is defined by
the equations  (\ref{eq:cve})
with the tangential variational equation (\ref{eq:tangenve})
removed as follows:
\begin{equation}
\label{eq:nveev}
\begin{array}{l}
  \dot{\eta'}_{j}=
  -4\mbox{sin}^{2}(\frac{j\pi}{2(n+1)})(\gamma_{2}+3\gamma_{4}
  cn^{2}(k;\alpha t))
  \xi'_{j},\\
  \dot{\xi'}_{j}=\eta'_{j}
\end{array}
\end{equation}
for \(1\leq j(\ne \frac{n+1}{2})
  \leq n\).

 Let us  consider the
monodromy matrices \(g\)
 defined by the analytic continuation of the solution
\(\zeta'(t)=(\xi'_{1}(t),\eta'_{1}(t),\cdots,\hat{\xi}'_{\frac{n+1}{2}}(t),
\hat{\eta}'_{\frac{n+1}{2}}(t),\cdots,\xi'_{n}(t),\eta'_{n}(t))\)
of the NVE (\ref{eq:nveev})
along the  periodic orbits in the phase curves \(\Gamma(\epsilon,t)\)
 as follows:
\begin{equation}
\label{eq:monodef4}
   \mbox{\boldmath$\zeta'$}(T_{1})=g_{1}\mbox{\boldmath$\zeta'$}(0),\quad
   \mbox{\boldmath$\zeta'$}(T_{2})=g_{2}\mbox{\boldmath$\zeta'$}(0).
\end{equation}
 The periods in Eqs. (\ref{eq:monodef4}) are \(T_{1},T_{2}\) in
Eqs. (\ref{eq:rel2}),
 respectively.
 These two fundamental periods \(T_{1}\) and \(T_{2}\) naturally
 form the parallelogram, whose associate monodromy matrices are given by
 \(g_{1}g_{2}g^{-1}_{1}g^{-1}_{2}(\equiv g_{*})\).
 These monodromy matrices are naturally endowed
 with the symplectic structure and the
 pairing properties of the eigenvalues, namely
 \(\{\sigma_{1},\sigma^{-1}_{1},\sigma_{2},\sigma^{-1}_{2},
    \cdots,\sigma_{n},\sigma^{-1}_{n}\}\).
 In general, the explicit calculation of the eigenvalues of
 the monodromy matrices is an unsuccessful business
 except rare cases such as Hamiltonian systems with
 homogeneous polynomial functions\cite{yo}, {\it Riemann's equation}, and the
 {\it Jordan-Pochhammer equations}\cite{3takano,3hejhal,3churchill}.
 This is one of the
 unavoidable difficulties
 in performing Ziglin's analysis of general dynamical systems.
 In our case, however, we can compute explicitly
 the eigenvalues of
 the {\it commutator} \(g_{*}=g_{1}g_{2}g^{-1}_{1}g^{-1}_{2}\)
 by using the fact that
the normal variational equation (\ref{eq:nveev})
 happens  to be in a class of
 {\it Lam\'e
 equations} \cite{whit}
\begin{equation}
 \frac{d^{2} y}{dt^{2}}-(E_{1}sn^{2}(k;\alpha t)+E_{2})y=0,
\end{equation}
 where \(E_{1}\) and \(E_{2}\) are constants, and the eigenvalues \(\sigma\)
 of the commutator \(g_{*}=g_{1}g_{2}g^{-1}_{1}g^{-1}_{2}\) are known
 \cite{whit} to be
  determined by the
 indicial equation
\begin{equation}
\label{eq:indicial1}
    \Delta^{2}-\Delta-(\alpha^{2}k^{2})E_{1}=0,
    \quad \sigma=\mbox{exp}(2\pi i\Delta)
\end{equation}
 with the singular point (pole) \(\tau\) located at the center of
  the parallelogram:
\begin{equation}
\tau=\frac{T_{1}+T_{2}}{2}=\frac{2K(k)+iK'(k)}{\alpha}.
\end{equation}
 If  we apply
 the indicial equation(\ref{eq:indicial1}) to the
normal variational equations (\ref{eq:nveev})
of the FPU lattices, the exponents of
the eigenvalues of \(g_{*}\)
are given by
\begin{equation}
  \Delta^{2}-\Delta-12\frac{\gamma_{4}}{\alpha^{2}k^{2}}
  \mbox{sin}^{2}(\frac{j\pi}{2(n+1)})=0.
\end{equation}
Noting
\begin{equation}
  \frac{\gamma_{4}}{\alpha^{2}k^{2}}=1
\end{equation}
from (\ref{eq:3rel1}), we finally obtain the eigenvalues of the commutator as
\begin{eqnarray}
\label{eq:fpumonodromy}
  \sigma_{j}=\mbox{exp}(2\pi i \frac{1\pm\sqrt{1+
  48\mbox{sin}^{2}\frac{j\pi}{2(n+1)}}}{2})\nonumber\\
	    =-\mbox{exp}(\pm \pi i\sqrt{25-24\mbox{cos}(\frac{j\pi}{n+1})})
\end{eqnarray}
for \(1\leq j(\ne \frac{n+1}{2})\leq n\), because there is {\it only one}
pole singularity inside the parallelogram formed by counterclockwise
closed loop of the monodromy group \(g_{1}g_{2}g^{-1}_{1}g^{-1}_{2}\).
 Even for more general nonlinear
lattices  given by
 potential functions (\ref{eq:potential}) of degree \(2m\)
 we can also compute the phase factors.
Let us consider singular solutions
  \(\phi(t)\) and \(\xi_{j}(t)\) whose singularities are located
 at \(t=\tau\) in the complex time plane as follows:
 \begin{equation}
 \label{eq:assumpsyg}
 \begin{array}{l}
   \phi(t)=C'(t-\tau)^{\beta},\quad \beta<0\nonumber\\
   \xi_{j}(t)=C''(t-\tau)^{\nu_{j}},
 \end{array}
 \end{equation}
 where \(C'\) is a particular constant, \(C''\) is an arbitrary constant.
 Thus, we can easily obtaine the formulas  the relations
 for \(\beta,C',\nu_{j}\) as
\begin{equation}
\begin{array}{l}
  \beta=\frac{-1}{m-1},\\
  (C')^{2m-2}=-\frac{\beta(\beta-1)}{2\gamma_{2m}},\\
  \nu_{j}^{2}-\nu_{j}-2\frac{m(2m-1)}{(m-1)^{2}}\mbox{sin}^{2}(\frac{j\pi}
  {2(n+1)})=0
\end{array}
\end{equation}
by the equations of motion.
 With the use of the fact that  the phase factor
 is obtained by means of the analytic continuation along the
 closed loop around the singular point \(t^{\nu_{j}}\), we obtain
 the phase factors as
\begin{equation}
\label{eq:generalmonodromy}
\begin{array}{c}
 \mbox{exp}(2\pi i \nu_{j})\\
 =\mbox{exp}\{2\pi i \frac{1\pm \sqrt{1+8\frac{m(2m-1)}{(m-1)^{2}}
 \mbox{sin}^{2}(\frac{j\pi}{2(n+1)}}}{2})\}\\
 =-\mbox{exp}\{\pm \pi i \sqrt{1+8\frac{m(2m-1)}{(m-1)^{2}}
 \mbox{sin}^{2}(\frac{j\pi}{2(n+1)})}\}
\end{array}
\end{equation}
 where \(2m\) is the degree of the potential polynomial  of \(
 \upsilon(X)\) in (\ref{eq:potential}). It is easy to confirm that in
 the case that \(2m=4\),
the  formula (\ref{eq:generalmonodromy}) recover the eigenvalues
(\ref{eq:fpumonodromy}) of the monodromy matrices
obtained from the {\it Jacobi elliptic function}. However, in the other
case that \(2m>4\),
we cannot compute the eigenvalues of monodromy matrices only from the
phase factors in general,
because the normal variational equations do not belong
to a class of {\it Lam\'e equations}.
To summarize, we can say
that
it is essential
to perform the singularity analysis towards non-homogeneous
 nonlinear lattices using the explicit eigenvalues
of the monodromy matrices that
the special solutions can  be written in terms of elliptic functions and
the normal variational equations are in a type of {\it Lam\'e equations}.

\section{Non-integrability theorem}
\subsection{Non-integrability proof in the integrable limit}
\setcounter{equation}{0}
The chief purpose of the present section is to give a theorem
telling the non-integrability by using the monodromy matrices obtained
in Section 2. We consider  monodromy matrices \(g\).
If the eigenvalues \(\{\sigma_{1},\sigma^{-1}_{1},\cdots,
 \sigma_{n},\sigma^{-1}_{n}\}\)
of monodromy matrices \(g\) do not satisfy the following relation
\begin{equation}
  \sigma_{1}^{l_{1}}\sigma_{2}^{l_{2}}\cdots \sigma_{n}^{l_{n}}=1
\end{equation}
 for any set of integers \(\{l_{1},\cdots,l_{n}\}\) except the trivial
case \(l_{1}=l_{2}=\cdots =l_{n}=0\), we call the monodromy matrices \(g\)
{\it non-resonant}.
It is already known that
the existence of a non-resonant monodromy matrix is a basic
assumption in order to perform Ziglin's analysis\cite{zig}.
Moreover, if there are {\it straight-line solutions}
such as (\ref{eq:compsol}) whose monodromy matrices
are non-resonant and if the variational equations can be diagonalized into
a decoupled form like (\ref{eq:cve}),
  Ziglin's theorem can be generalized to
Yoshida's theorem for Hamiltonian systems composed by
kinetic energy terms and potential energy terms\cite{yopd}.

Yoshida's theorem
asserts the following in terms of two different monodromy matrices
\(\{g_{a},g_{b}\}\):
Suppose that there exists an additional complex analytic integral, which
is holomorphic along the solution (\ref{eq:solsol1}), and that
one of the monodromy matrices
\(g_{a}\) is non-resonant. Then it is necessary that
 one of the following  two cases, namely,
\begin{itemize}
 \setlength{\itemsep}{0pt}
 \item[(I)]
\(g_{b}(\lambda_{j})\)  must preserve the eigendirection of \(g_{a}(\lambda_{j})\)
, i.e., \(g_{a}(\lambda_{j})\) must commute with \(g_{b}(\lambda_{j})\),
\item[(II)]
\(g_{b}(\lambda_{j})\)
 must permute the eigendirection of \(g_{a}(\lambda_{j})\),
 i.e., \(g_{b}(\lambda_{j})\) is written by
 \(\left[
 \begin{array}{cc}
   0 & \beta \\
   -\frac{1}{\beta} & 0
 \end{array}
 \right]\) in the base of \(g_{a}\)
 having the
 eigenvalues \(i\) and \(-i\)
\end{itemize}
 for some suffix \(j\),
at least occurs
for any other  monodromy matrix
\(g_{b}\) represented in the
basis of
 \(g_{a}\).
As for the non-resonance condition of the monodromy
matrix \(g_{*}=g_{1}g_{2}g^{-1}_{1}g^{-1}_{2}\), the following
lemma is already known:
\newtheorem{th2}{Lemma}
\begin{th2}[\cite{4um0,4umth},1994]
\label{th3:fputheorem1}
The n quantities
\(\{\sqrt{25-24\mbox{cos}\frac{j\pi}{n+1}}| j=1,\cdots,n\}\)
are rationally independent.
\end{th2}

We remark that
   the rational independency of the set\\
\(\{\sqrt{25-24\mbox{cos}\frac{j\pi}{n+1}}|j=1,\cdots n\}\)
guarantees that
 the commutator \(g_{*}=g_{1}g_{2}g^{-1}_{1}g^{-1}_{2}\) whose
eigenvalues are given in Eq. (\ref{eq:fpumonodromy}) is non-resonant.

For connection with the  algebraic number theory on the {\it cyclotomic field}
\(\mbox{\boldmath$Q$}(\mbox{exp}(\frac{\pi i}{n+1}))\) over
\(\mbox{\boldmath$Q$}\) see Ref. \cite{4umth}.
Thus, we can regard that the non-resonance hypothesis holds for
the monodromy matrix  \(g_{*}\).
Now, using Yoshida's theorem and the variational
analysis in the former section, we obtain a theorem which proves
the non-integrability of a FPU lattice for an arbitrary set of
the system parameters
\(\{n,\mu_{2},\mu_{4}|n(\geq3):\mbox{odd},\mu_{2}>0,\mu_{4}>0\}\) as follows:
\newtheorem{th3}{Theorem}

\begin{th3}
\label{th3:fputheorem}
A FPU lattice (\ref{eq:ulam}) which is characterized by
 an arbitrary set of the system parameters
\(\{n,\mu_{2},\mu_{4}|n \geq3, n:\mbox{odd},\mu_{2}>0,\mu_{4}>0\}\)
 has no  analytic first
integrals besides the Hamiltonian itself
for fully small energy \(\epsilon(\approx 0)\).
\end{th3}

({\bf Proof of Theorem \ref{th3:fputheorem1}})

  From the non-resonance hypothesis on \(g_{*}\),
  we can apply   the Yoshida theorem\cite{yopd} to
  the FPU lattices.
   According to the above argument, if we prove that at least one
 monodromy matrix \(g_{s}\in\{g_{1},g_{2}\}\)
does not have the following properties

\begin{itemize}
 \setlength{\itemsep}{0pt}
\item[(a)] \(g_{s}(\lambda_{j})\)
  preserves the eigendirection
 of \(g_{*}(\lambda_{j})\)
\\and
\item[(b)] \(g_{s}(\lambda_{j})\)
 permutes the eigendirection of \(g_{*}(\lambda_{j})\)(the eigenvalues of
 \(g_{s}(\lambda_{j})\) are \(i,-i\)),
\end{itemize}
 at once
 for any suffix \(j\)(\(1\leq j(\ne n+1/2) \leq n\)) of \(g_{s}(\lambda_{j})\),
 then the
 FPU lattices
 \cite{fpu} are concluded to  have
 no other analytic conserved quantities besides the Hamiltonian
 itself, i.e., the assertion of the present theorem holds.
In the following, we will show that \(g_{1}(\lambda_{j})\) for any \(j(\ne
\frac{n+1}{2})\) has neither the property (a) and the property (b).

When we take the
 limit
 \(\epsilon\rightarrow 0\), the relations
\begin{equation}
\gamma_{4}=\mu_{4} C^{2},\cdots,\gamma_{2m}=
  \mu_{2m}C^{2m-2}\rightarrow 0,\quad \alpha
 \rightarrow \sqrt{2\mu_{2}},\quad
 k\rightarrow 0
\end{equation}
hold
  and we can compute the periods of the monodromy matrices
 \(g_{1},g_{2}\) in the limit as
\begin{equation}
 T_{1}\rightarrow \frac{1}{\sqrt{2\gamma_{2}}}\pi,\quad
 T_{2}\rightarrow \frac{1}{\sqrt{2\gamma_{2}}}\pi+i\infty
\end{equation}
 by  the formula (\ref{eq:rel2}).
To compute  the  monodromy matrix \(g_{1}(\lambda_{j})\) in this
limit, we rewrite
the variational equations (\ref{eq:nveev}) in terms of
the modulus of the elliptic integral \(k\) as follows
\begin{equation}
\label{eq:simvar}
  \ddot{\xi'}_{j}+
  4\mu_{2}\mbox{sin}^{2}(\frac{j\pi}{2(n+1)})(1+\frac{2k^{2}}{1-2k^{2}}
   cn^{2}(k;\alpha t))
  \xi'_{j}=0,\quad j=1,\cdots,n
\end{equation}
by virtue of equations (\ref{eq:cdterm}) and (\ref{eq:3rel1}).
For arbitrary \(k\in\left[0,1\right]\),
we have a fundamental system of solutions as
\(\{\xi^{a}_{j}(k;t),\xi^{b}_{j}(k;t)\}\) of (\ref{eq:simvar}) satisfying
\begin{equation}
\label{eq:initcs}
\begin{array}{l}
\xi^{a}_{j}(k;0)=1,\quad \frac{d}{dt}\xi^{a}_{j}(k;t)|_{t=0}=
\dot\xi^{a}_{j}(k;0)=0\\
\xi^{b}_{j}(k;0)=0,\quad \frac{d}{dt}\xi^{b}_{j}(k;t)|_{t=0}=
\dot\xi^{b}_{j}(k;0)=1.
\end{array}
\end{equation}

Because \(cn^{2}(k;\alpha t)\) in Eqs. (\ref{eq:simvar})
has the following Taylor expansions \cite{byrd} at
\(\kappa \equiv k^{2}=0\):
\begin{equation}
  cn^{2}(k;\alpha t) = \mbox{cos}^{2}(\frac{\alpha\pi}{2K} t)
   -\frac{1}{2}\kappa \mbox{sin}^{2}(\frac{\alpha\pi}{2K} t)
    \mbox{cos}^{2}(\frac{\alpha\pi}{2K} t)+\cdots
\end{equation}
and analytic in \(\kappa\) at \(\kappa=0\), the
fundamental system of solutions \(\{\xi^{a},\xi^{b}\}\) are also analytic

 in \(\kappa\) at \(\kappa=0\) \cite{ito}  as
\begin{equation}
\begin{array}{l}
  \xi^{a}_{j}(k;t)=\xi^{a}_{j,0}(t)+\xi^{a}_{j,1}(t)\kappa
  +\xi^{a}_{j,2}(t)\kappa^{2}+\cdots\\
  \xi^{b}_{j}(k;t)=\xi^{b}_{j,0}(t)+\xi^{b}_{j,1}(t)\kappa
+\xi^{a}_{j,2}(t)\kappa^{2}+\cdots,
\end{array}
\end{equation}
 where the unperturbed parts
\(\{\xi^{a}_{j,0},\xi^{b}_{j,0}\}\) are obtained by
\begin{equation}
\label{eq:fusy}
\begin{array}{l}
 \xi^{a}_{j,0}(t)=\mbox{cos}
 (2\mbox{sin}(\frac{j\pi}{2(n+1)})\sqrt{\mu_{2}}t),\\
  \xi^{b}_{j,0}(t)=\frac{1}{2\sqrt{\mu_{2}}\mbox{sin}(\frac{j\pi}{2(n+1)})}
\mbox{sin}(2\mbox{sin}(\frac{j\pi}{2(n+1)})\sqrt{\mu_{2}}t)
\end{array}
\end{equation}
 from  the initial conditions (\ref{eq:initcs}).
The monodromy matrix \(g_{1}(\lambda_{j})\) is given by
\begin{equation}
  g_{1}(\lambda_{j})=\left[
  \begin{array}{cc}
  \xi^{a}_{j}(k;T_{1}) & \xi^{b}_{j}(k;T_{1})
 \\
\dot\xi^{a}_{j}(k;T_{1}) & \dot\xi^{b}_{j}(k;T_{1})
  \end{array}\right],
\end{equation}
 and
with the use of the fact that
the fundamental system of solutions (\ref{eq:fusy})
are analytic in \(\kappa\) at \(\kappa=0\), we show that
 \(g_{1}(\lambda_{j})\) for \(j=1,\cdots, n\) have
 also the Taylor expansions as
\begin{equation}
g_{1}(\lambda_{j})=g_{1,0}(\lambda_{j})+g_{1,1}(\lambda_{j})\kappa
  +g_{1,2}(\lambda_{j})\kappa^{2}+\cdots.
\end{equation}
Thus now, when we consider
the low energy limit (\(\epsilon\rightarrow 0\), \(\kappa\rightarrow 0\)),
 we  obtain the formula
\begin{equation}
  g_{1}(\lambda_{j})\rightarrow\left[
  \begin{array}{cc}
  \mbox{cos}(\sqrt{2}\pi\mbox{sin}(\frac{j\pi}{2(n+1)})) &
\frac{1}{2\mbox{sin}(\frac{j\pi}{2(n+1)})\sqrt{\mu_{2}}}
 \mbox{sin}(\sqrt{2}\pi\mbox{sin}(\frac{j\pi}{2(n+1)}))
 \\
 -2\mbox{sin}(\frac{j\pi}{2(n+1)})\sqrt{\mu_{2}}
 \mbox{sin}(\sqrt{2}\pi\mbox{sin}(\frac{j\pi}{2(n+1)}))
 &
\mbox{cos}(\sqrt{2}\pi\mbox{sin}(\frac{j\pi}{2(n+1)})\pi)
  \end{array}\right].
\end{equation}
This means that
the eigenvalues of \(g_{1}(\lambda_{j})\) tend to
\begin{equation}
\label{eq:limiteigenvalues}
\{\mbox{exp}(i\pi(\sqrt{2}\mbox{sin}(\frac{j\pi}{2(n+1)})),
\mbox{exp}(-i\pi(\sqrt{2}\mbox{sin}(\frac{j\pi}{2(n+1)}))
\}
\end{equation}
 and
 \(g_{1}(\lambda_{j})\) for any \(j(\ne \frac{n+1}{2})\)
 does not have the property of (b).\\

Now assume that
 \(g_{2}(\lambda_{j})\) for
  some
 \(j(\ne\frac{n+1}{2})\) has
 the property of (a). Then if \(g_{1}(\lambda_{j})\)
 as well as \(g_{2}(\lambda_{j})\) has also the
 property of (a), we have
\begin{equation}
\label{eq:commono}
 g_{*}(\lambda_{j})=g_{1}(\lambda_{j})g_{2}(\lambda_{j})
 g^{-1}_{1}(\lambda_{j})g^{-1}_{2}(\lambda_{j})=
 \mbox{\boldmath$id$},
\end{equation}
 where \(\mbox{\boldmath$id$}\) denotes
 the \(2\times 2\) identity matrix. This relation (\ref{eq:commono})
  means that \(g_{1}(\lambda_{j})\)
 and \(g_{2}(\lambda_{j})\) commute each other and clear contradicts
 the non-resonance hypothesis of \(g_{*}\).
Consider the other case where
  \(g_{1}(\lambda_{j})\) has the property of (a)
  and \(g_{2}(\lambda_{j})\) does not have the property of (a).
  However, in the representation of \(g_{1}(\lambda_{j})\) and
 \(g_{2}(\lambda_{j})\) in the basis of \(g_{*}(\lambda_{j})\) as
\begin{equation}
  g_{1}(\lambda_{j})=\left[
  \begin{array}{cc}
  \mu & 0 \\
   0 & \frac{1}{\mu}
  \end{array}\right], \quad
  g_{2}(\lambda_{j})=\left[
  \begin{array}{cc}
   a & b \\
   c & d
  \end{array}\right]\mbox{  }(ad-bc=1),
\end{equation}
  the following relation
\begin{equation}
\label{eq:kankei1}
  g_{*}(\lambda_{j})=g_{1}(\lambda_{j})g_{2}(\lambda_{j})
  g_{1}^{-1}(\lambda_{j})g_{2}^{-1}(\lambda_{j})=\left[
  \begin{array}{cc}
  ad-\mu^{2}bc & ab(\mu^{2}-1) \\
  cd(\frac{1}{\mu^{2}}-1) & ad -\frac{bc}{\mu^{2}}
  \end{array}\right]
\end{equation}
  must be satisfied. Since \(g_{*}(\lambda_{j})\) is assumed to have a
  diagonal representation as
  \(g_{*}(\lambda_{j})=\mbox{diag}\left[
  \sigma_{j},\sigma_{j}^{-1}\right]\) and from the relation
(\ref{eq:kankei1}),
  we obtain
\begin{equation}
  a=0,\quad d=0,\quad bc=-1,
\end{equation}
  when  \(g_{*}(\lambda_{j})\ne
  \mbox{\boldmath$id$}\);  \(g_{2}(\lambda_{j})\)
must have the property of (b).

 Therefore, in  the basis of \(g_{*}(\lambda_{j})\),
 we have
\begin{equation}
\label{eq:basedia}
  g_{1}(\lambda_{j})=\left[
  \begin{array}{cc}
  \mu & 0 \\
  0 & \frac{1}{\mu}
  \end{array}\right] \mbox{  and} \quad
  g_{2}(\lambda_{j})=\left[
  \begin{array}{cc}
  0 & \beta \\
  -\frac{1}{\beta} & 0
   \end{array}\right].
\end{equation}
These relations (\ref{eq:basedia}) result in
\begin{equation}
\label{eq:comrel}
 \left[
 \begin{array}{cc}
 \sigma_{j} & 0 \\
 0 & \sigma^{-1}_{j}
 \end{array}\right]=
 g_{*}(\lambda_{j})=g_{1}(\lambda_{j})
g_{2}(\lambda_{j})g_{1}^{-1}(\lambda_{j})
 g_{2}^{-1}(\lambda_{j})=g_{1}^{2}(\lambda_{j}),
\end{equation}
where
\(\sigma_{j}=
-\mbox{exp}\{\pi i\sqrt{25-
24\mbox{cos}\frac{j\pi}{n+1}}\}\).
However, the relation (\ref{eq:comrel})
causes again a contradiction  with the fact that
the eigenvalues of \(g^{2}_{1}(\lambda_{j})\) approach to
\begin{equation}
\{\mbox{exp}(i\pi(2\sqrt{2}\mbox{sin}(\frac{j\pi}{2(n+1)})),
\mbox{exp}(-i\pi(2\sqrt{2}\mbox{sin}(\frac{j\pi}{2(n+1)}))
\}
\end{equation}
in the limit \(\epsilon\rightarrow 0\) and   there is a difference between
the eigenvalues of \(g_{*}(\lambda_{j})\) and the eigenvalues of
\(g^{2}_{1}(\lambda_{j})\) as will be  shown in
the Appendix.

We have seen that \(g_{1}(\lambda_{j})\) for any
\(j(\ne \frac{n+1}{2}) \) has neither
the property of (a) or the property of (b).
Now the theorem holds.
(\hbox{\rule{6pt}{6pt}} {\bf End of proof of Theorem \ref{th3:fputheorem1}})\\

\section{Classification  of non-homogeneous nonlinear lattices}
\setcounter{equation}{0}
 Here in the present section, we consider the classification of
these non-integrable nonlinear lattices via the degree of non-integrability.
We define the degree of non-integrability  as follows:
\newtheorem{df}{Definition}
\begin{df}
We write a relation
\begin{equation}
  \{H,\epsilon\} \sim \{H',\epsilon'\}
\end{equation}
 if and only if
 Hamiltonian systems  with \(n\) degrees of freedom \(H\) and \(H'\)
 have the same number of additional analytic integrals which are functionally
independent together with the Hamiltonians \(H\) and \(H'\) respectively.
Here \(\epsilon(\epsilon')\) denotes  the total energy of the
Hamiltonian systems \(H(H')\).
\end{df}
 If we associate the positive integer
\begin{equation}
\varrho =
2^{r}\cdot 3^{n-r-1}
\end{equation}
with a Hamiltonian system \(H\)
with \(n\) degrees of
freedom which has \(r\) additional analytic integrals which are functionally
independent together with the Hamiltonian \(H\), we can classify
Hamiltonian systems via the degree of non-integrability as follows:
\begin{equation}
  \{H,\epsilon\}\sim
  \{H',\epsilon'\} \iff \varrho(\{H,\epsilon\})=\varrho(\{H',\epsilon'\}).
\end{equation}
 For a non-homogeneous nonlinear lattice
\begin{equation}
\label{eq:nnnl2m}
  H_{\mu_{2},\mu_{4},\cdots,\mu_{2m}}(\mbox{\boldmath$q,p$};t)
=\frac{1}{2}\sum_{i=1}^{n}p^{2}_{i}+\sum_{k=1}^{m}\frac{\mu_{2k}}{2k}
   \sum_{i=1}^{n}(q_{i-1}-q_{i})^{2k}
\end{equation}
 of degree \(2m\),
consider  the following transformation \(\Re \) from
  a non-homogeneous nonlinear lattice to
  another  non-homogeneous nonlinear lattice:
\begin{df}
 We call a transformation \(\Re \) from
  \(H_{\mu_{2},\mu_{4},\cdots,\mu_{2m}}(\mbox{\boldmath$q,p$};t)\) to
  \(H_{\mu'_{2},\mu'_{4},\cdots,\mu'_{2m}}(\mbox{\boldmath$q',p'$};t')\)
  a {\bf homogenizer}
   if the relations
\begin{equation}
\begin{array}{l}
\label{eq:homogenizer}
  t'=
 {\alpha}t,
  \quad\mbox{\boldmath$q'$}= \frac{1}{\alpha^{\frac{1}{m-1}}}
  \mbox{\boldmath$q$},
  \quad\mbox{\boldmath$p'$}=
  \frac{1}{\alpha^{\frac{m}{m-1}}}\mbox{\boldmath$p$},\\
  \mu'_{2}=\frac{1}{\alpha^{2}}\mu_{2},\quad
   \mu'_{4}=\frac{1}{\alpha^{2\frac{m-2}{m-1}}} \mu_{4},\quad \cdots,\quad
   \mu'_{2\nu}=\frac{1}{\alpha^{2\frac{m-\nu}{m-1}}}, \quad\cdots,\quad
   \mu'_{2m}=\mu_{2m},
\end{array}
\end{equation} are satisfied
 where \(\alpha\) is a real quantity larger than unity \((1<\alpha <\infty)\).
\end{df}

{\it Remark 1}

The transformation \(\Re\) in Eqs. (\ref{eq:homogenizer})
 can be regarded as a generalization of
the scaling transformation for homogeneous nonlinear lattices
\(H_{\mu_{2}=0,\cdots,\mu_{2m-2}=0,\mu_{2m}}\).
Moreover, the equations of motion
itself remain invariant under the transformation \(\Re\),
 while a Hamiltonian is changed according to
 the variations  of coupling constants
\(\{\mu_{2\gamma}|1 \leq \gamma <m\}\). Thus, we can say that \(\Re\)
preserves the degree of non-integrability as
\begin{equation}
   \{ \Re H,\Re \epsilon \} \sim \{ H,\epsilon\}.
\end{equation}

 {\it Remark 2}

 The operation \(\Re\) corresponds to a kind of
{\it renormalization group} along
 the time axis from the view point
of the coarse graining in statistical physics
 and it is easy to see that the fixed point of \(\Re\)
  is the homogeneous nonlinear lattice \(H^{*}\), i.e.,
\begin{equation}
   H^{*}=\Re H^{*},
\end{equation}
  where
\begin{equation}
\label{eq:fixh}
\begin{array}{l}
\label{eq:hhhh}
   H^{*}=\lim_{l \rightarrow \infty} \Re^{l} H\\
        =
  \frac{1}{2}\sum_{i=1}^{n}p^{2}_{i}+\frac{\mu_{2m}}{2m}
   \sum_{i=1}^{n}(q_{i-1}-q_{i})^{2m}
\end{array}
\end{equation}
 and \(\Re^{l}\) denotes  \(l\) iterations of the
operation \(\Re\).
This is the reason why the \(\Re\) is called a {\it homogenizer} here.\\

{\it Remark 3}

Under the transformation \(\Re\), the total energy \(\epsilon\) is changed
into \(\epsilon'\) by the formula
\begin{equation}
 \epsilon'= \Re \epsilon =\frac{1}{\alpha^{\frac{2m}{m-1}}}\epsilon.
\end{equation}

By the above remarks,
we can conclude that
\begin{equation}
\label{eq:nhh}
  \{H_{\frac{1}{\alpha^{2l}}\mu_{2},\mu_{4}},\frac{1}{\alpha^{4l}}\epsilon\}
  \sim
  \{H_{\mu_{2},\mu_{4}},\epsilon\}
\end{equation}
for an arbitrary FPU lattice \(\{H_{\mu_{2},\mu_{4}},\epsilon\}\).
If we take the limit
 \(\alpha^{l}\rightarrow \infty\)
 preserving the relation \(\frac{\epsilon}{\alpha^{4l}}\approx O(1)\),
 L.H.S. of (\ref{eq:nhh}) approaches a homogeneous
nonlinear lattice of degree 4, which means that
the  degree of non-integrability of a homogeneous nonlinear lattice is
 the
same as the degree of non-integrability of a FPU lattice in the high energy
limit \(\epsilon\rightarrow\infty\). In relation to the non-integrability
of the homogeneous nonlinear lattice, there is a known result
by Yoshida\cite{yo} stating
as follows:\\

Considering \(n\) number
of distinct exponents
\begin{equation}
\label{eq:3sin2}
\Delta_{j}\equiv
\sqrt{1+\frac{8m(2m-1)}{(m-1)^{2}}\mbox{sin}^{2}(\frac{j\pi}{2(n+1)})}
 \quad(\mbox{for}\quad j=1,\cdots,n),
\end{equation}
 or the \(n\) representative of the {\it Kowalevski exponents}\cite{3yo1} of
 the homogeneous nonlinear lattice of degree \(2m\).
If the n quantities \(\{\Delta_{j}\}\) in (\ref{eq:3sin2})
are rationally independent,
then the
homogeneous nonlinear lattices  have no  analytic first
integrals except the Hamiltonian itself for \(2m(\geq 4)\) and
for odd \(n(\geq 3)\).\\

Note  that the \(n\) quantities (\ref{eq:3sin2}) equal the phase factors
(\ref{eq:generalmonodromy}) of the eigenvalues of the commutator \(g_{*}\).
Thus, the non-resonance hypothesis
proving the non-integrability in the
 low energy limit \(\epsilon \rightarrow 0\)
induces also the non-integrability
of nonlinear lattices with a nonhomogeneous potential function
in the high energy limit \(\epsilon \rightarrow \infty\).
The reason for this uniqueness of the non-resonance hypothesis
lies in the fact that
after a conformal mapping \(z=\phi^{4}(t)\),
the closed loop of the commutator \(g_{1}g_{2}g^{-1}_{1}
g^{-1}_{2}\) becomes a homotopy  equivalent loop which corresponds
to Yoshida's construction of the monodromy matrix \(G=(G_{0}G_{1})^{16}\)
giving the same non-resonance hypothesis in terms of Kowalevski exponents.
Here, \(G_{0}\) corresponds to
a counterclockwise
closed loop around the singularity \(z=0\) and \(G_{1}\) corresponds to
another counterclockwise  closed loop around the singularity \(z=1\) in the complex \(z\) plane, where
\(z\) satisfies the following Gauss hyper geometric equation:
\begin{equation}
  z(1-z)\frac{d^{2}\xi'_{j}}{dz^{2}}+\left[\frac{3}{4}-\frac{5}{4}z\right]
 \frac{d\xi'}{dz}+\frac{3}{2}\gamma_{4}\mbox{sin}^{2}(\frac{j\pi}{2(n+1)})
 \xi'_{j}=0
\end{equation}
into
which the conformal mapping \(z=\phi^{4}\) transforms the variational
 Eq. (\ref{eq:cve}) with
the condition \(\gamma_{2}=0\) and \(G_{0},G_{1}\) are the two
fundamental monodromy groups of the Gauss hyper geometric equation.
See also Fig. 3.\\

So far, we have succeeded in checking the non-integrability of the FPU
lattices both in the low energy limit and in the high energy limit from a
single non-resonance hypothesis. However, there still remain some
problems about the integrability of FPU lattices for a finite
energy \(0<\epsilon <\infty\). To consider the problem, we introduce a
real parameter
\begin{equation}
  \chi=\frac{\epsilon \mu_{4}}{(n+1)\mu^{2}_{2}}>0,
\end{equation}
where \(\chi\) is found to be a dimensionless parameter, because
\begin{equation}
  \mbox{Dimension of }\chi=
  \frac{\left[\frac{\mbox{L}^{2}}{\mbox{MT}^{2}}\right]
  \left[\frac{1}{\mbox{MT}^{2}\mbox{L}^{2}}\right]}
{\left[1\right]\left[\frac{1}
  {\mbox{M}^{2}\mbox{T}^{4}}\right]}=\left[1\right]
\end{equation}
by the standard unit system as
\(\left[\mbox{M}\right]=\left[\mbox{kg}\right],
\left[\mbox{T}\right]=\left[\mbox{sec}\right],
\left[\mbox{L}\right]=\left[\mbox{m}\right]\).
 We remark here that \(\chi\) is invariant under the transformation
 \(\Re\) preserving the degree of non-integrability. This means that
it is sufficient for us to classify the degree of non-integrability
of the FPU lattices by using
this  dimensionless parameter \(\chi\).
Furthermore, with the use of  the parameter \(\chi\),
we can replace the formula of the modulus
of the elliptic function \(k\)
 in Eq. (\ref{eq:modulus}) by
\begin{equation}
k=\frac{1}{\sqrt{2}}
\sqrt{1-\frac{1}{\sqrt{1+4\chi}}}.
\end{equation}
 We note here that \(k\rightarrow 0\) if and only if
 \(\chi \rightarrow 0\) and this
 limit can be realized for each set of \(\{\epsilon,\mu_{2},\mu_{4}|
 0<\epsilon,\mu_{2},\mu_{4}<\infty\}\) when we take \(n\)(:odd) a fully
great number. If we recall that in Theorem 1
it is essential
for proving the
non-integrability of the FPU lattices
to take the limit \(k\rightarrow 0(\chi\rightarrow 0)\)
so that
the eigenvalues of \(g_{1}(\lambda_{j})\) approach the quantities
in Eq. (\ref{eq:limiteigenvalues}),
 we have the following theorem:

\begin{th3}
\label{th3:fpufinal}
A
FPU lattice(\ref{eq:ulam}) characterized by
 an arbitrary set of the system parameters
\(\{\epsilon,\mu_{2},\mu_{4}|0<\epsilon<\infty, 0<\mu_{2}<\infty,
0<\mu_{4}<\infty\}\)
 has no additional  analytic
integrals of motion besides the Hamiltonian itself for
a fully great  number of degrees of freedom \(n(n:\mbox{odd})\).
\end{th3}

This result shows that we need
a fully great number \(n\) of degrees of freedom to guarantee the strongest
non-integrability  \(\varrho(\{H_{FPU},\epsilon\})=3^{n-1}\)
 of each FPU lattice \(H_{FPU}\) with the systems parameters
\(\{\mu_{2},\mu_{4}|0<\mu_{2}<\infty,0<\mu_{4}<\infty\}\) for
\(0<\epsilon<\infty\) and a problem remains still open as to
 whether we can prove
 the non-integrability
 of each FPU lattices with the system parameters
\(\{\mu_{2},\mu_{4},n|0<\mu_{2}<\infty,0<\mu_{4}<\infty,
n:\mbox{odd},n\geq 3\}\)
for a given finite energy \(\epsilon(0<\epsilon<\infty)\).
Clearly, Theorem 1 and Theorem 2 have the reciprocal conditions
in terms of
the system parameters \(\{\epsilon,\mu_{2},\mu_{4},n\}\)
and we can
unify them into a single theorem as follows:
\begin{th3}
A FPU lattice(\ref{eq:ulam}) characterized by the system parameters\\
\(\{\epsilon,\mu_{2},\mu_{4},n|0<\epsilon<\infty,0<\mu_{2}<\infty,
0<\mu_{4}<\infty,
n\geq 3, n:\mbox{odd}
\}\) has no additional analytic integrals of motion besides the
Hamiltonian itself when the following condition
\begin{equation}
\chi=\frac{\epsilon \mu_{4}}{(n+1)\mu^{2}_{2}}\rightarrow 0
\end{equation}
is satisfied.

\end{th3}
\section{Summary and discussion}
We have got the non-integrability proof
 of the FPU lattices
 by tracing the following steps.
In Section 2, we have shown that the singularity analysis can be performed towards
 the non-homogeneous nonlinear lattices by finding the special
straight-line solutions in terms of elliptic functions.
Especially, it is shown there that the normal variational equations
can be decoupled into  separated equations
and each of them happens to be
a  Lam\'e equation and consequently  we can compute
the eigenvalues of the monodromy
matrices associated with the counterclockwise loop along the
period cell of the special solutions.
  In  Section 3, with the use of Yoshida's theorem whose  validity is
guaranteed when
that the normal variational equations can be decoupled,
 we can prove the non-existence of the additional analytic conserved
quantities
besides the Hamiltonian itself
for  FPU lattices in the low energy limit.
In Section 4,
we have considered the classification of FPU lattices via the
  degree of non-integrability. There, by introducing  a transformation
\(\Re\) from a non-homogeneous nonlinear lattice
into another  non-homogeneous nonlinear lattice,
which preserves the equations of motion, we have shown that the degree of
non-integrability of each
FPU lattice in the high energy limit is the same as that
 of the homogeneous nonlinear lattice whose non-integrability
is known to be proven by the analysis
using  Kowalevski exponents.
In case of FPU lattices with  arbitrary energy in \(0<\epsilon<\infty\),
 we have a theorem showing  the non-integrability for a fully great
number of degrees of freedom
by introducing a dimensionless parameter which characterizes the
normal variational equations. This theorem (Theorem 2) and Theorem 1
have the reciprocal sufficient conditions for the non-integrability of
FPU lattices and finally they are  unified into a single theorem (Theorem 3)
 by the  dimensionless parameter \(\chi\).

There remains an open problem to be solved:
Can we have a theorem about the non-integrability of more general
nonlinear lattices of degree \(2m>4\)?  More specifically speaking, can
  we have a theorem asserting the non-integrability of the
non-homogeneous nonlinear lattices of degree \(2m>4\) such as Theorem 1 and
Theorem 2?
The present neck is in a fact that
the normal variational equations along the special solutions
of {\it hyper-elliptic functions}
do not belong to a class of Lam\'e equations,
though they can always be decoupled into \(n-1\)
separated variational equations,
and
we cannot compute the eigenvalues of the
monodromy matrices associated with these variational equations
in general  and therefore we cannot perform
 the Ziglin analysis completely.
 However, the  problem on the degree of
non-integrability for  more general
non-homogeneous nonlinear lattices remains an interesting
open question to be solved,  because
the present analysis on the non-integrability of the FPU lattices,
together with Yoshida's argument about the homogeneous
nonlinear lattice in terms of  Kowalevski exponents, strongly suggests
 universality about the non-integrability
of more general non-homogeneous nonlinear lattices in Eq.
(\ref{eq:nnnl2m}).\\

\section{Appendix}
\setcounter{equation}{0}
In this appendix, we show the difference between the eigenvalues of
  \(g_{*}(\lambda_{j})\) and the eigenvalues of \(g^{2}_{1}(\lambda_{j})\)
for any \(j(\ne\frac{n+1}{2})\).
We prove
this by reductio ad absurdum. First, we  assume that the following equality
\begin{equation}
\label{eq:heq}
\begin{array}{l}
  \mbox{Spec} (g^{2}_{1}(\lambda_{j}))\equiv
\{\mbox{exp}(i\pi(2\sqrt{2}\mbox{sin}(\frac{j\pi}{2(n+1)})),
\mbox{exp}(-i\pi(2\sqrt{2}\mbox{sin}(\frac{j\pi}{2(n+1)}))
\}\\
=
\{-\mbox{exp}(\pi i\sqrt{25-
24\mbox{cos}\frac{j\pi}{n+1}}),
-\mbox{exp}(-\pi i\sqrt{25-
24\mbox{cos}\frac{j\pi}{n+1}})\}
\equiv \mbox{Spec}(g_{*}(\lambda_{j}))

\end{array}
\end{equation}
would hold for some \(j(\ne\frac{n+1}{2})\).
Our purpose here is to show that the relation (\ref{eq:heq}) is false.
By (\ref{eq:heq}),
 one of the following equalities
\begin{equation}
\label{eq:eqp}
\sqrt{25-
24\mbox{cos}\frac{j\pi}{n+1}}+1+2m_{+}=
2\sqrt{2}\mbox{sin}(\frac{j\pi}{2(n+1)})
\end{equation}
\begin{equation}
\label{eq:eqm}
\sqrt{25-
24\mbox{cos}\frac{j\pi}{n+1}}+1+2m_{-}=
-2\sqrt{2}\mbox{sin}(\frac{j\pi}{2(n+1)})
\end{equation}
 holds for \(m_{+},m_{-}\in \mbox{\boldmath$Z$}\) and
  \(j\ne\frac{n+1}{2}\).
Because of the inequalities
\begin{equation}
1<\sqrt{25-
24\mbox{cos}\frac{j\pi}{n+1}}<7,\quad
0<2\sqrt{2}\mbox{sin}(\frac{j\pi}{2(n+1)})<2\sqrt{2},
\end{equation}
it is sufficient to consider the equalities (\ref{eq:eqp}) and
(\ref{eq:eqm}) for
\(m_{+}\in \{-3,-2,-1,0\}\) and \(m_{-}\in \{-5,-4,-3,-2\}\).
If we rewrite \(\mbox{sin}(\frac{j\pi}{2(n+1)})\) as \(X\),
 Eq.
(\ref{eq:eqp}) and Eq. (\ref{eq:eqm})
are changed into the following equations
\begin{equation}
40X_{+}^{2}+(1+2m_{+})4\sqrt{2}X_{+}+1-(1+2m_{+})^{2}=0,
\end{equation}
\begin{equation}
40X_{-}^{2}-(1+2m_{-})4\sqrt{2}X_{-}+1-(1+2m_{-})^{2}=0
\end{equation}
respectively.
By considering \(X(=X_{+},X_{-}\geq 0\),
we can easily obtain a series of  the solutions as
follows:
\begin{equation}
\begin{array}{l}
X_{+}(m_{+}=0)=0\\
X_{+}(m_{+}=-1)=\frac{\sqrt{2}}{10}\\
X_{+}(m_{+}=-2)=\frac{1}{\sqrt{2}}\\
X_{+}(m_{+}=-3)=\frac{5\sqrt{2}+\sqrt{290}}{20}>\frac{7+\sqrt{289}}{20}=
  \frac{6}{5}>1\\
X_{-}(m_{-}=-2)=\frac{\sqrt{2}}{5}\\
X_{-}(m_{-}=-3)=\frac{-5\sqrt{2}+\sqrt{290}}{10}\\
X_{-}(m_{-}=-4)=\frac{1}{\sqrt{2}}\\
X_{-}(m_{-}=-5)=\frac{-9\sqrt{2}+\sqrt{962}}{20}.
\end{array}
\end{equation}
Furthermore, because of the following relations
\(0<X_{\pm}<1,\quad X_{\pm}\ne \frac{1}{\sqrt{2}}\) for
\(j \ne \frac{n+1}{2}\),
we can discard the solutions \(X_{+}(m_{+}=0),X_{+}(m_{+}=-2),X_{+}(m_{+}=-3),
X_{-}(m_{-}=-4)\). For the solutions
\(X_{+}(m_{+}=-1),X_{-}(m_{-}=-2)\), we have the equalities as
\begin{equation}
\label{eq:relap1}
\begin{array}{l}
\mbox{cos}\frac{j\pi}{n+1}=1-2(X_{+}(m_{+}=-1))^{2}=\frac{24}{25}\\
\mbox{cos}\frac{j\pi}{n+1}=1-2(X_{-}(m_{-}=-2))^{2}=\frac{21}{25}
\end{array}
\end{equation}
respectively. However,
these relations (\ref{eq:relap1}) contradict with a fact
\cite{4um0}  that
\begin{equation}
\mbox{cos}\frac{j\pi}{n+1}\in \mbox{\boldmath$Q$} \iff
  \mbox{cos}\frac{j\pi}{n+1}=0,\quad
\mbox{or}\quad \frac{1}{2},\quad \mbox{or}\quad -\frac{1}{2}.
\end{equation}
Thus, we can also discard the solutions \(X_{+}(m_{+}=-1)\) and
\(X_{-}(m_{-}=-2)\).
Let us consider the solutions \(X_{-}(m_{-}=-3)\) and \(X_{-}(m_{-}=-5)\).
If we rewrite \(\mbox{exp}\frac{j\pi}{n+1}\) as \(\zeta\),
the following relation
\begin{equation}
\label{eq:relap2}
  25\zeta^{4}+70\zeta^{3}-46\zeta^{2}+70\zeta+25=0
\end{equation}
is  satisfied for the solution \(X_{-}(m_{-}=-3)\),
while  the relation
\begin{equation}
\label{eq:relap3}
  25\zeta^{4}+462\zeta^{3}+626\zeta^{2}+462\zeta+25=0
\end{equation}
is satisfied for the solution \(X_{-}(m_{-}=-5)\).
 Since \(\zeta\) is one of \(2(n+1)\)-th root of unity and
 \(2(n+1)\quad (\mbox{mod} \quad 4)=0\), the minimal polynomial \(P(\zeta)\)
with  coefficients of
rational integers which has a solution \(\zeta\) for \(P(\zeta)=0\)
must be one of the {\it cyclotomic polynomials} \(\Phi_{l}(Y)\in
\mbox{\boldmath$Z$}[Y]\):
\begin{equation}
  \Phi_{l}(Y)=\prod_{i=1}^{\varphi(l)}(Y-\zeta_{i}),
\end{equation}
 where \(2(n+1) (\mbox{ mod }l) =0\),
\(\{\zeta_{i}\}\) is  a set of all primitive \(l\) th roots of
unity and \(\varphi(l)\) is the number of positive integers which are less
than or equal to \(l\) and relatively prime to \(l\) and this
function \(\varphi\) is called
Euler's \(\varphi\)-function.
All cyclotomic polynomials \(\Phi_{l}(Y)\) are known to be
irreducible over \(\mbox{\boldmath$Q$}\) and we can check that
 cyclotomic polynomials
\(\Phi_{l}(Y)\) whose degrees are less than or equal to \(4\)
 are restricted
to the case\cite{4umth} that
\begin{equation}
\begin{array}{l}
 \aleph \equiv \{ \Phi_{l}(Y)|2\leq\mbox{Deg}\Phi_{l}(Y)\leq 4\}\\ =
\{\Phi_{3}(Y)=Y^{2}+Y+1,\Phi_{4}(Y)=Y^{2}+1,
 \Phi_{5}(Y)=Y^{4}+Y^{3}+Y^{2}+Y+1,\\
\Phi_{6}(Y)= Y^{2}-Y+1,
 \Phi_{8}(Y)=Y^{4}+1,
 \Phi_{12}(Y)=Y^{4}-Y^{2}+1\}.
\end{array}
\end{equation}
Using this fact, we can say  that  a cyclotomic polynomial
\(\Phi_{l}(Y)\in \aleph\) must divide either
the  polynomial
\begin{equation}
  P_{1}(Y)\equiv 25Y^{4}+70Y^{3}-46Y^{2}+70Y+25
\end{equation}
in Eq.(\ref{eq:relap2})
 or the  polynomial
\begin{equation}
  P_{2}(Y)\equiv 25Y^{4}+462Y^{3}+626Y^{2}+462Y+25
\end{equation}
in
Eq.(\ref{eq:relap3}). It is easy to check that
this is impossible.  Thus now, we can
conclude that the assumption (\ref{eq:heq}) is false; i.e., we have
shown that
for any \(j(\ne \frac{n+1}{2})\)
\begin{equation}
  \mbox{Spec}(g_{1}^{2}(\lambda_{j}))\ne \mbox{Spec}(g_{*}(\lambda_{j})).
\end{equation}\\

\Large
{\bf Acknowledgements}
\normalsize\\

The majority of this work was carried out to complete
the doctoral dissertation at Department of Physics,
University of Tokyo.
 I would
like to thank
Prof. Masuo Suzuki, Prof. Miki Wadati and Prof. Haruo Yoshida for
 valuable discussions.
 While writing this work up
I have  been supported
   from
 the Special Researcher's Program towards  Basic
  Science at the RIKEN
 and from the Program
 of the Complex Systems II at the International Institute for
 Advanced Study (IIAS-Kyoto). I would like to thank Prof. Shun-ichi
Amari for his
continual encouragement.

\clearpage

redpajama_set_name: RedPajamaArXiv


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Over 26,000 new projects related to blockchain were created last year on GitHub, a code source website according to data composed by risk and financial advisory company Deloitte.
The report, titled, "Evolution of Blockchain Technology: Insights from the GitHub Platform" states that the site is home to more than 86,000 blockchain projects, counting main initiatives like bitcoin.
Another important takeaway from the piece was the fact that a predominate amount of blockchain ventures are mostly sedentary on the site.
"The stark reality of open-source projects is that most are abandoned or do not achieve meaningful scale. Unfortunately, blockchain is not immune to this reality. Our analysis found that only 8 percent of projects are active, which we define as being updated at least once in the last six months," stated Deloitte.
San Francisco has the most blockchain projects with over 1200 users and 101 organizations.
London came in second place with 919 ventures overall, trailed by New York with 774.
"It is also worth noting the high level of activity in China, specifically, Shanghai and Beijing. In both of these cities, most of the projects pertain to cryptocurrencies and cryptocurrency exchanges, with an emphasis on scalability," according to Deloitte.


A special report (January, 2016) by Wikistrat, an online global strategic consultancy, on China's strategic calculus in North Korea explains Beijing's ambivalent response to North Korea's nuclear attics.
The report recommends that the United States should coordinate greater coercive pressure on North Korea among regional stakeholders, including China, South Korea, Japan and other Western allies.
What is missing is the strategic use of economic development to transform the behavior of an essentially insecure regime. What comes to mind is the trajectory of Vietnam from a belligerent Communist hotbed to a modern economic dynamo and a growing ally with the United States. A recent report on North Korea apparently building special economic zones including transport infrastructure should give some food for thought.
A report of The Economist Intelligence Unit underscore three trends that will define businesses in 2016 - parsimony, regulatory tightening, and digital disruption.
"....... the trends set to play out in 2016 will not just cause companies problems, but also spur fresh opportunities. Spreading public health provision in developing countries, for instance, will open up new markets for medicines and hospital care. Nimble financial technology ("FinTech") companies intend to run rings around the giants of their industry—many of whom, though belaboured with regulations, nonetheless remain mighty. Their interest margins stand to be bolstered by rising interest rates in the US and UK; despite tough economic conditions, many of their emerging market peers are in strong positions to weather any financial storms.
The onslaught of internet access and usage is throwing the world of telecoms companies upside down, but also affording them chances to open up new streams of revenue: providing content directly to consumers, for instance. Companies in all industries will strive to collect, interpret and capitalise on vast amounts of new data. "Big data" will be harnessed to understand consumers, improve healthcare and cut firms' energy bills. The pan-industry push towards more environmentally friendly use of energy will spur everything from production of renewable power to the spread of industrial automation and the "internet of things".
As the wheels of business whir on, in the background the 12 signatories to the Trans-Pacific Partnership trade agreement will go about the slow process of domestically ratifying the deal. That will be difficult, but is likely to happen eventually—just not in 2016".
How to accumulate wealth for your family in the 21st century, with apologies to Mother Teresa & Thomas Piketty is my PowerPoint presentation, capturing the essence of Thomas Piketty's 685-page opus Capital in the 21st Century, (The Belknap Press of Harvard University Press, Cambridge, Massachusetts and London, England, 2014) and re-interpreting it in the family wealth accumulation context.


text:
using QuantConnect.Interfaces;
using QuantConnect.Packets;

package com.quantconnect.lean.Commands
{
    /**
     * Represents a command that will change the algorithm's status
    */
    public class AlgorithmStatusCommand : ICommand
    {
        /**
         * Gets or sets the algorithm status
        */
        public AlgorithmStatus Status { get; set; }

        /**
         * Initializes a new instance of the <see cref="AlgorithmStatusCommand"/>
        */
        public AlgorithmStatusCommand() {
            Status = AlgorithmStatus.Running;
        }

        /**
         * Initializes a new instance of the <see cref="AlgorithmStatusCommand"/> with
         * the specified status
        */
        public AlgorithmStatusCommand(AlgorithmStatus status) {
            Status = status;
        }

        /**
         * Sets the algoritm's status to <see cref="Status"/>
        */
         * @param algorithm The algorithm to run this command against
        public CommandResultPacket Run(IAlgorithm algorithm) {
            algorithm.Status = Status;
            return new CommandResultPacket(this, true);
        }
    }
}


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I hope everyone fights you. That would be a good challenge.
But that was era 20, I will not make any claims or promises but I intend on having a good era.
I hope all players within Zetamania will look at this era as a new beginning and not hold a grudge for past wars.
I would simply like to say that playing with the KD of Ferakin was quite the honor. Some really good players in there.
P.S. I really hate this map.
Why do you say that Karta?
One blocker and two strategically placed walls and that whole corner is secured. The only little problem being needing slaves fr the walls, but that shouldn't take long after OOP war.
hell its possible with NO walls to block off this area.
Well, 2 walls and 1 blocker OR 4 blockers and 1 wall. You at least need 1 wall to block between the mountans and the edge of the map. I'll be happy to show you how to block it off with only 2 walls and a blockers... after we're done taking it all. Unless Senturu or anyone else in your KD wants to explain it to you.
but here goes. Build a city on the bridges. also build a city in the south to block that point. up on the northern part. 2 well places blockers will nearly block off access from the north. and yea a wall or a city of adaquate size should be able to fill that hole.
thats what i can see anyway.
I apologize for creating a second "Zetamania 21," for I did not notice this thread.
BL, could you announce the name of this traitorous scum for all to see?
I am now convinced that Death Dealers are not a KD worthy of any consideration whatsoever, as it is become apparent they rely on traitor-tactics. Should the Death Dealers restore their honor, I suggest the traitor's cities be recaptured by the old KD or B&D by conquerors.
That then, is my treatise.
I do hope that if any honorable players exist within DD, that one of them succinctly betray their KD in retaliation, in punishment, of the excessively despicable actions. Perhaps this player would serve the victimized KD.
I hereby state my rejection of any excuses that pardon such actions under various guises, especially concerning gameplay in VU.
May the scum ridden coward face emasculation.
Situational comparison doesn't alter the reality of the situation. Whatever Mr. Rylee did, I leave you to go address that on your own. Besides, betrayal is different than city-feeding. But we are not here to debate past eras or Mr. Rylee. It's fairly obvious that betraying your own Kingdom is worse than tricking the system to feed cities to your Kingdom.
Either way, it is clear that Mr. Dandelion cannot be trusted by any KD at all. But hey, if DD needs to scrape the bottom of the bucket for scum in order to accomplish a victory this era, then perhaps we should all have a good drink about it and laugh at their pseudo-achievements.
Karta was in death dealers since the era start (I think) because he is the reason why I joined DD.
And I'm pretty sure Dandelion started at DD's core and joined because he was pretty much surrounded (Like me XD).
Dandelion should not have joined DeathDealers while he had perfect access to the KoF core. He singlehandedly gave it a nice wrecking. Had it been a simple matter of being surrounded, he ought to have honorably fought things through. There is no dishonor in being defeated by a powerful enemy, but there is dishonor in being a traitor.
You, Yukan, did not KD switch mid-era. Therefore, you aren't guilty of jacking stuff up outside the proper bounds. However, DD has quite the asset where you are concerned, as you know just what you are doing.
If that is the case, perhaps my observations are due for revision. From what I know, at this point, it was a kind of betrayal.
Why delete your character when you could simply leave KD? Either way, I would quite enjoy playing alongside you again at some point.
Rylee did no such thing, he did not want to be a kingdom mage as designated to you when you pick Elf in our kingdom.
Perhaps BL can comment on who left his kingdom, I didn't see anybody change banners.
I will explain - he's talking about the general conversation where the name Dandelion has been repeatedly mentioned.
I would like everyone to send out scouts. I want to know where every KD is located asap. We have not found any other KD yet so we do not need many GTs atm; however, build at least 500 GTs in all your cities anyway. I would also like all our orcs to settle a OOP city in KoF core when we find them. Those orcs should build 2-4k GTs and 1-2k MTs in the city they build in KoF's core, then start spamming troops in their core. We will bring the fight to them this age/era.
I asked him to join because it was the start of the age and he was near our core. We did not see/find KoF until sometime later. Thus, if he left you guys it was way before we had any knowledge if him leaving you guys or anything related to that matter.
Now my question is why are we only hearing about this now almost a week after he joined us?
I left Ferakin last era, after spending an era where they won without any contribution from me, to find some challenges on my own. I landed without a kingdom in the north-east corner, with the other solo players. Instead of playing on my own, I decided accept some offers to join DD, despite the provocation of Bidi (who will be on my to-kill-list from now on lolol).
Then my analysis is incorrect.
I'm afraid you joined the wrong KD if you're looking for a challenge.
2: who cares about these "tactics" things like this happen. KoF was to Cocky and we didnt get to destroying the DD scouts/citys as fast as we should and are paying the price for it.
3: i dont know what else i was going to post here, and i dont want to read the previous 18 comments to remember.
So... how's the war going for everyone?
I'm terribly sorry Mr. Dandelion. But we have lost the battle.
We killed or injured a total of 307 enemy troops.
Also, Dandelion didn't make anyone anything. You were always going to be our enemy, even from the start. It was always going to be Red vs. Blue. The Reds just weren't ready for the Blues to take the fight to them so soon in the era! We aren't going to have a repeat of last era. You may well beat me and Dandelion and retake your cities (the ones we don't destroy). But all this time you waste on us, the rest of our Kingdom grows stronger. We are just the first wave, the vanguards, the foerunners of your demise. So go ahead and kill my armies, more will take my place, and then I will return again to finish what was started!
Beautifully written, this will make your crushing defeat ever more glorious for the KINGDOM OF FERAKIN!
Mr. Dandilion spawned beside me and was kingdomless. I invited him to join Deathdealers. I'm glad our honor has been restored. I agree those that kingdom jump during protection should be looked down upon!
I just realized that DD has an 8 player advantage against the second largest KD, lol.


text:
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The Tea Act was an act of the Parliament of Great Britain. The main idea was to reduce the large amount of tea the British East Indian Company obtained and to assist them in the company survival.
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The wheels of progress are turning faster than ever, with Zynga announcing the imminent closure of three games, launched as recently as December 2012. CityVille 2, Party Place and The Friend Game are all now facing the axe, as part of Zynga's "disciplined" approach to portfolio management.
CityVille 2 missed our expectations despite pushing the envelope in 3D gaming and both The Friend Game and Party Place failed to achieve the mass market appeal Zynga franchise are known for.
CityVille 2 enjoyed a healthy-enough launch in November, but interest dropped off quickly during December, apparently paired with a decrease in support from the studio. A similar thing happened with The Ville, which launched strongly, dropped drastically, but is still available on Facebook, alongside the original CityVille.
An interesting element of these closures is tied in with recent Zynga staff changes. Brian Reynolds, Chief Game Designer, left the company just last week – and it was his studio that oversaw development on CityVille 2. Conspiracy theorist gamers are wondering if the two are somehow connected.
Alternatively, it could just be the biggest example yet of Zynga's ruthless new cost-cutting measures. The company closed down 13 titles last quarter alone, suggesting a new approach: Perform or perish.