A Framework for Theory Development in Design Science Research: Multiple Perspectives Abstract It has taken more than 350 years, from Francis Bacon (1994 Bacon ( /1620 to Karl Popper (1989) , to define and refine the Western scientific notion of "theory". Shelves of books (Dubin, 1978 , is an example well known to IS Ph.D. students) and reams of academic publications have been written in an attempt to define theory and evolve methods for its creation. Gregor (2006) recently published an already widely cited paper just on the single aspect of identifying types of theory in IS. We give this prelude to provide perspective on the limitations of any single paper in the area of theory development. Our goal is not to define theorizing in design science research in IS (DSRIS) but rather to move a step toward that goal by describing a framework for theory development in DSRIS and providing some simple illustrations of its use. We wish to point out that the emphasis of the post-hoc theory derivations we provide above is on exercising the DSRIS theory development framework more than on the specific theoretical statements we derive. We realize that the nature of hermeneutic explication and the brief space we have for the explanation of our reasoning may make the specifics of our DREPT statements questionable to some. However, it should be noted that, through the use of our framework, we produce logically sound theoretical statements with clearly defined dependent and independent variables and obvious design implications. And this is precisely the sort of testable-through-artifactconstruction theory we hoped our framework would enable. An example of use of an early version of the framework in an actual DSRIS project is developed in Kuechler and Vaishnavi (2008b) . We present the kernel theory and DREPT propositions from that project in Appendix A.. In the course of developing a framework for theory development in DSRIS and illustrating its use, we necessarily touched on a wide variety of topics in limited depth. Working from a generalized notion of mid-range theory as applied to DSRIS, we expand on Gregor's (2006) typology of theory in DSRIS to include both theory for design and action (ISDT: Type V theory) and a design-relevant mid-range explanatory/predictive theory: DREPT. From that expansion, we propose a hierarchy of theory in DSRIS arranged according to level of abstraction of theoretical constructs. We further explicate the framework by an epistemological perspective that stresses the framework principle that transitioning between theory levels from kernel theory to artifact corresponds to increasing specialization (concretization) of independent and dependent variables. The framework by itself is useful for understanding design theories, their relations to kernel theories, and the roles each plays in the process of design. When combined with traditional theory development techniques -ways of thinking about design for design theory development (Appendix C) -the framework is useful in directing the construction of design theory as well.. We exercise our framework using two published examples of DSRIS. For both examples, we use our framework to formally explicate the ISDT that each paper developed. We additionally develop a DREPT for each to demonstrate the potential that that type of theory can have for DSRIS. As a midrange theory, each DREPT is generalizable to the design of any artifact utilizing the phenomena of its grounding kernel theory(s) to achieve artifact effects.. In the first of the examples, Kasper (1996) , the transition from kernel theories in psychology to an ISDT for a DSS exhibiting improved calibration seemed straightforward; however, closer examination showed that there was no logical reason or published empirical basis for the mapping from kernel theory constructs to ISDT constructs. We developed post-hoc a sophisticated and potentially very valuable DREPT to make that mapping logic explicit. We believe a broad gap between kernel theory constructs and design features is not unusual and that it increases with the amount of intuition required by the researcher to reason from kernel theory to ISDT.. In our second example, Arnott (2006) , the gap from kernel theories to ISDT was substantially less; the mapping from kernel theory constructs to ISDT constructs was more transparent. However, by combining the techniques of induction and triangulation of perspectives, we were able to develop a DREPT that made explicit the substantial amount of information that was produced in Arnott's DSRIS project yet remained tacit. By reflecting on the theories that underlie the artifact and by using induction on the data produced by the artifact evaluation effort, we are able to propose a DREPT with very broad application -almost any IS development project -and correspondingly significant potential value.. As mentioned previously, this paper encompasses diverse material. We believe the breadth is required to ground the novel aspects of our framework within the context of theory development in DSRIS, which is relatively new itself. In much of the seminal DSRIS literature, theorizing beyond design models (ISDT) is conspicuously absent in descriptions of the outputs from design science research. Some of the reasons for this are historical and are described in Kuechler and Vaishnavi (2008a) ; significantly, many prominent design science researchers in IS were trained in computer science and engineering disciplines, and carried over a very pragmatic focus on the artifact-ascontribution. It is only in some of the newer publications on DSRIS (Gregor & Jones, 2007; Kuechler, Park, & Vaishnavi, 2009; Kuechler & Vaishnavi, 2008b; Venable, 2006 ) that explanatory/predictive theory has been mentioned as a possible contribution from a DSRIS project. We try to motivate the development of DREPT within DSRIS from two standpoints: (1) as a new but potentially valuable means of capturing design knowledge that would otherwise remain tacit in the design artifact and process and (2) as a formalism in our DSRIS theory development framework that helps to explain the nature of design theory and its relationship to both the artifact and kernel theories. William James once described philosophy as "an unusually stubborn attempt to think clearly and consistently" (Putnam, 1997) . We hope to persuade the field that theory development in DSRIS be considered not as a complication or distraction, but rather as "an unusually stubborn attempt to capture knowledge clearly and consistently for the benefit of practice and cumulative research efforts".

O B O B I O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O B I I O O O O O O O O O O O O O O O B O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O B I O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O B I O O O B O O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


TeraStitcher

A

tool

for

fast

automatic

of

teravoxel-sized

microscopy

images

Background:

Further

advances

in

modern

microscopy

are

leading

to

teravoxel-sized

tiled

images

at

high

resolution,

thus

increasing

the

dimension

of

the

stitching

problem

of

at

least

two

orders

of

magnitude.

The

existing

software

solutions

do

not

seem

adequate

to

address

the

additional

requirements

arising

from

these

datasets,

such

as

the

minimization

of

memory

usage

and

the

need

to

process

just

a

small

portion

of

data.

Results:

We

propose

a

free

and

fully

automated

Stitching

tool

designed

to

match

the

special

requirements

coming

out

of

teravoxel-sized

tiled

microscopy

images

that

is

able

to

stitch

them

in

a

reasonable

time

even

on

workstations

with

limited

resources.

The

tool

was

tested

on

teravoxel-sized

whole

mouse

brain

images

with

micrometer

resolution

and

it

was

also

compared

with

the

state-of-the-art

stitching

tools

on

megavoxel-sized

publicy

available

datasets.

This

comparison

confirmed

that

the

solutions

we

adopted

are

suited

for

stitching

very

large

images

and

also

perform

well

on

datasets

with

different

characteristics.

Indeed,

some

of

the

algorithms

embedded

in

other

stitching

tools

could

be

easily

integrated

in

our

framework

if

they

turned

out

to

be

more

effective

on

other

classes

of

images.

To

this

purpose,

we

designed

a

software

architecture

which

separates

the

strategies

that

use

efficiently

memory

resources

from

the

algorithms

which

may

depend

on

the

characteristics

of

the

acquired

images.

Conclusions:

TeraStitcher

is

a

free

tool

that

enables

the

stitching

of

Teravoxel-sized

tiled

microscopy

images

even

on

workstations

with

relatively

limited

resources

of

memory

GB)

and

processing

power.

It

exploits

the

knowledge

of

approximate

tile

positions

and

uses

ad-hoc

strategies

and

algorithms

designed

for

such

very

large

datasets.

The

produced

images

can

be

saved

into

a

multiresolution

representation

to

be

efficiently

retrieved

and

processed.

We

provide

TeraStitcher

both

as

standalone

application

and

as

plugin

of

the

free

software

Forthcoming

microscopy

techniques

like

Confocal

Light

Sheet

Microscopy

are

able

to

acquire

tiled

images

of

teravoxel

or

more.

These

images

pose

novel

requirements

to

stitching

that

are

not

adequately

addressed

by

existing

tools..

In

this

paper

we

have

presented

a

tool

designed

for

automatic

stitching

of

teravoxel-sized

tiled

images.

The

tool,

initially

developed

for

stitching

images

generated

by

the

CLSM

microscope,

is

completely

general,

and

suited

to

be

adapted

to

other

instruments

capable

to

acquire

images

of

these

dimensions.

The

central

idea

is

to

accurately

specify

the

requirements

of

the

stitching

problem

when

teravoxel-sized

images

are

involved,

and

then

use

efficiently

the

system

resources

to

perform

the

stitching

of

these

images.

To

improve

the

generality

of

the

proposed

tool,

we

defined

a

software

architecture

that

clearly

separates

the

strategies

in

common

among

different

datasets

from

the

algorithms

that

may

depend

on

specific

characteristics

of

the

acquired

images..

An

implementation

of

the

tool

that

is

capable

to

perform

the

stitching

of

real

teravoxel-sized

images

on

workstations

with

relatively

limited

memory

resources

in

reasonable

time

has

been

presented.

It

uses

specific

algorithms

well

suited

for

a

relatively

wide

class

of

images,

that

substantially

reduce

memory

and

computational

requirements

of

some

basic

steps

of

the

stitching

procedure

with

respect

to

previous

approaches.

A

comparison

with

state-of-the-art

stitching

tools

confirms

that

the

solutions

In

this

table

we

report

a

comparison

of

execution

times

(in

seconds)

and

memory

peaks

(in

MB)

between

TeraStitcher

and

the

state-of-the

art

stitching

tools

when

stitching

Megavoxel-sized

datasets.

a

Time

spent

in

stitching,

net

of

I/O.

O O O O O O O O O O B O O O O O O B O O O O O B I O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O B I O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O B O O O O O O B O O O


Proposal

for

a

multilevel

university

cybermetric

analysis

model

Abstract

Universities'

online

seats

have

gradually

become

complex

systems

of

dynamic

information

where

all

their

institutions

and

services

are

linked

and

potentially

accessible.

These

online

seats

now

constitute

a

central

node

around

which

universities

construct

and

document

their

main

activities

and

services.

This

information

can

be

quantitative

measured

by

cybermetric

techniques

in

order

to

design

university

web

rankings,

taking

the

university

as

a

global

reference

unit.

However,

previous

research

into

web

subunits

shows

that

it

is

possible

to

carry

out

systemic

web

analyses,

which

open

up

the

possibility

of

carrying

out

studies

which

address

university

diversity,

necessary

for

both

describing

the

university

in

greater

detail

and

for

establishing

comparable

ranking

units.

To

address

this

issue,

a

multilevel

university

cybermetric

analysis

model

is

proposed,

based

on

parts

(core

and

satellite),

levels

(institutional

and

external)

and

sublevels

(contour

and

internal),

providing

a

deeper

analysis

of

institutions.

Finally

the

model

is

integrated

into

another

which

is

independent

of

the

technique

used,

and

applied

by

analysing

Harvard

University

as

an

example

of

use.

From

the

outset,

the

Internet

and

the

World

Wide

Web

have

been

very

closely

linked

to

universities,

both

in

their

design,

development

and

expansion,

and

in

the

impact

of

these

services

on

the

way

the

institution

works

(Goldfarb

Castells

Berners-Lee

and

Fischetti

The

accessibility,

affordability

and

personalisation

of

these

information

and

communication

technologies

(ICT)

are

factors

in

their

penetration

and

impact

on

all

walks

of

life;

higher

education

institutions

are

particularly

influenced

by

four

disruptive

forces

(Katz

-Unbundling.

Content

producers

and

creators

can

deconstruct

services,

allowing

greater

flexibility

in

services

on

offer.

-Demand-Pull.

Users

can

assemble

content

to

suit

their

own

interests.

-Ubiquitous

access.

Online

content

is

accessible

from

a

multitude

of

devices,

without

geographic

or

temporal

restrictions.

-Intellectual

property.

There

is

a

need

for

real

and

effective

rules

that

describe

how

individuals

enter

and

exit

the

online

university

community..

Since

universities

are

organisations

that

depend

on

attracting,

developing

and

organising

human

talent

for

the

purpose

of

creating

and

disseminating

intellectual

capital

(Katz

some

of

the

activities

they

should

consider

with

respect

to

the

impact

of

Internet

use

are

(Katz

-Develop

new

points

of

view

on

research

processes,

educational

content,

publications,

software,

and

informational

instruments

and

resources.

-Find

out

how

new

communication

tools

(blogs,

wikis,

mashups,

and

many

more)

will

interact

with

traditional

academic

approaches

and

thinking..

PREPRINT.

Please

cite

as:

Orduña-Malea,

E.,

Proposal

for

a

multilevel

university

cybermetric

analysis

model.

Scientometrics,

v.

doi:

-Understand

what

data

and

information

the

organisation

is

responsible

for,

and

how

they

will

be

stored

and

protected.

-Understand

what

influence

the

institution's

digital

resources

can

have

on

its

reputation.

-Maintain

and

enhance

the

online

identity

of

the

institution.

The

proposed

analysis

model,

based

on

parts

(core

and

satellite),

levels

(institutional

and

external)

and

sublevels

(contour

and

internal),

is

simple,

independent

of

technique

(allows

constant

updating

of

the

model

without

having

to

modify

its

general

architecture),

and

provides

structured

information

which

enables

a

complete

analysis

of

each

institution..

Cybermetrics

(insofar

as

it

is

a

technique

through

which

the

conceptual

analysis

model

may

be

applied)

facilitates

the

measurement

of

the

same

indicators

at

different

levels

of

institutional

analysis

(i.e.

it

enables

systemic

analyses),

something

not

possible

with

other

quantitative

techniques

such

as

bibliometrics,

which

depend

on

the

assignation

of

the

institution

on

the

part

of

the

authors

of

the

scientific

contributions..

Furthermore,

the

localisation

of

lower

units

favours

the

identification

of

institutions

with

missions

and

objectives

that

are

not

only

scientific,

and

this

may

assist

in

the

process

of

profiling

certain

traits

related

to

university

diversity

and

describing

the

general

performance

of

the

university

with

greater

precision..

Notwithstanding,

the

study

of

university

web

units

(internal

level)

is

complex.

The

analysis

of

Complutense

University

of

Madrid

shows

an

excessive

casuistic

of

URL

syntaxes

and

redirections.

This

reflects

in

a

lower

web

performance,

mainly

due

to

a

lack

of

appropriate

web

management

policies.

That

is,

the

internal

analysis

can

be

used

not

only

to

measure

web

production

or

impact

but

also

to

detect

the

degree

of

maturity

of

the

university

structure

on

the

Web

(low

in

the

case

of

Complutense

University)..

The

complete

analysis

of

Harvard

University

has

proven

the

advantages

of

the

model.

The

core

internal

level

shows

as

some

subdomains

related

to

service

activities

achieve

great

amount

of

performance.

The

model

allows

the

identification

of

unequal

distribution

of

impact

(both

institutional

and

external)

regarding

the

universities'

missions.

Satellite

level

shows

that

the

activities

done

by

universities

outside

the

official

website

are

significant

and

measurable.

For

all

these

reasons,

could

be

asserted

that

"harvard.edu"

is

not

enough

to

analyse

"Harvard

University"

under

a

webometric

point

of

view,

and

that

the

proposed

model

helps

to

expand

and

complete

this

analysis

by

identifying

and

describing

the

structure

of

the

website

(that

reflects

the

structure

of

the

organization),

and

assessing

the

web

performance

of

each

element

identified..

PREPRINT.

Please

cite

as:

Orduña-Malea,

E.,

Proposal

for

a

multilevel

university

cybermetric

analysis

model.

Scientometrics,

v.

doi:

Despite

the

model's

advantages,

it

also

has

certain

limitations,

not

conceptual

but

practical:

-The

model

has

too

many

instrumental

limitations:

the

shortcomings

in

the

construction

of

the

online

seats,

the

inaccuracy

of

some

indicators

(URL

mentions

for

external

impact),

and

the

inaccessibility

of

certain

indicators

(e.g.

web

traffic

data)

mean

that

some

indicators

or

units

cannot

be

adequately

measured.

-An

excessive

diversity

and

randomness

has

been

detected

in

the

URL

syntax

of

the

internal

unit

seats

(the

aforementioned

example

of

Complutense

University

of

Madrid

can

be

logically

extended

to

the

general

university

web

space);

this

should

be

corrected

with

a

suitable

institutional

web

policy.

-All

the

levels

can

contain

multidomains

and

URL

redirections

(e.g.

"hbs.harvard.edu"

and

"hbs.edu"),

which

make

the

identification

of

universities

and

quantification

of

content

more

complicated.

-The

internal

institutional

level

of

the

model

is

only

applicable

to

a

specific

university

system,

in

which

the

university

structure

is

clearly

specified

(legally

or

not),

and

does

not

have

to

coincide

with

the

university

systems

of

other

countries.

-Cybermetric

method

does

not

permit

the

direct

capture

of

some

diversity

categories,

such

as

those

of

components

or

programming.

Other

categories

and

attributes

(such

as

climate

diversity,

thematic

areas,

etc.)

must

be

manually

and

independently

considered

for

each

university

and

unit..

The

next

steps

to

be

taken

in

this

research

project

are

the

thorough

and

comprehensive

application

of

the

model

into

a

complete

university

system,

and

the

analysis

of

the

aforementioned

limitations,

with

the

aim

of

determining

its

scope

and

applicability

for

wider

cybermetric

analysis.

The

development

of

this

model

is

rooted

in

the

theoretical

study

of

cybermetric

university

analysis

units.

It

first

identifies

the

concept

of

web

document

and

subsequently

proposes

a

classification

of

logical

units

necessary

in

establishing

the

model

of

analysis..

Following

this,

the

conceptual

model

of

systemic

multilevel

university

analysis

proposed

by

Orduña-Malea

is

used

as

a

reference

model

that

is

independent

of

the

technique

employed

and

which

takes

into

account

the

attributes

of

university

diversity

and

multidimensionality.

The

model

is

adapted

to

be

applied

through

cybermetric

techniques..

The

internal

and

external

levels

are

demonstrated

through

entities

belonging

to

the

Complutense

University

of

Madrid

(the

highest

ranked

Spanish

university

on

the

Ranking

Web),

due

to

this

institution

clearly

shows

the

diversity

of

activities

and

a

wider

casuistry

of

URL

syntaxes,

necessary

to

assess

both

the

need

for

internal

analysis

and

its

complexity

(both

technical

and

organizational).

The

data

were

manually

collated

in

January

and

one

URL

-from

each

found

casuistry-was

randomly

selected,

to

illustrate

the

diversity

of

URL

creation

process..

The

full

application

of

the

model

is

performed

by

analysing

Harvard

University

(the

highest

ranked

university

on

the

Ranking

Web).

For

that

purpose

both

core

and

satellite

URLs

are

collected,

and

measured

at

institutional

and

external

levels

(levels

and

other

elements

of

the

model

will

be

defined

and

explained

in

section

-The

core

contour

level

is

composed

by

"harvard.edu"

official

URL..

-The

core

internal

level

is

composed

only

by

third-level

sub-domains

("x.harvard.edu/x")

in

order

to

facilitate

the

test

analysis.

In

any

case,

the

measurement

of

third-level

sub-domains

implies

the

measurement

of

the

remaining

levels,

and

considering

that

subdirectories

have

methodological

problems

to

be

accurate

analysed,

the

selected

sample

is

considered

representative.

All

URLs

were

gathered

by

browsing

the

official

website,

along

May,

-The

satellite

levels

(both

contour

and

internal)

were

identified

through

the

Harvard

Social

service.

The

platforms

considered

are

the

following:

Academia,

Facebook,

Flickr,

Twitter,

and

Youtube.

The

selection

of

these

platforms

is

exploratory

and

only

meant

to

show

the

webometric

performance

of

some

Harvard

University

channels

on

some

social

applications.

-Institutional

measures

(both

core

and

satellite)

were

measured

by

count

page

indicator,

using

<site:>

command

on

Google.

-External

measures

(both

core

and

satellite)

were

measure

by

URL

mention

indicator,

using:

<"URL"

-site:harvard.edu>

command

on

Google

for

core,

and

<"URL"

-site:domain.tld>

for

satellite

(substituting

"domain.tld"

for

each

platform:

academia.edu,

twitter.com,

etc.).

O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Flow

structures

in

cerebral

aneurysms

This

document

is

made

available

in

accordance

with

publisher

policies.

Please

cite

only

the

published

version

using

the

reference

above.

Full

terms

of

use

are

available:

Mechanical

properties

of

blood

flow

are

commonly

correlated

to

a

wide

range

of

cardiovascular

diseases.

In

this

work

means

to

describe

and

characterise

the

flow

field

in

the

free-slip

and

no-slip

domains

are

discussed

in

the

context

of

cerebral

aneurysms,

reconstructed

from

in-vivo

medical

imaging.

The

approaches

rely

on

a

Taylor

series

expansion

of

the

velocity

field

to

first

order

terms

that

leads

to

a

system

of

ODEs,

the

solution

to

which

locally

describes

the

motion

of

the

flow.

On

performing

the

expansion

on

the

vessel

wall

using

the

wall

shear

stress,

the

critical

points

can

be

identified

and

the

near-wall

flow

field

parallel

to

the

wall

can

be

concisely

described

and

visualised.

Furthermore

the

near-wall

expansion

can

be

expressed

in

terms

of

relative

motion,

and

the

near-wall

convective

transport

normal

and

parallel

to

the

wall

can

be

accurately

derived

on

the

no-slip

domain.

Together,

these

approaches

give

a

viable

and

robust

means

to

identify

and

describe

fluid

mechanic

phenomena

both

qualitatively

and

quantitatively,

leading

to

feasible

practical

use

in

biomedical

applications.

From

analysis

of

steady

and

unsteady

flow

simulations

in

two

anatomically

accurate

cerebral

saccular

aneurysm

cases,

a

set

of

measures

can

be

readily

obtained

at

all

time

intervals,

including

the

impingement

region,

separation

lines,

convective

transport

near

the

wall

and

vortex

core

lines

or

structures,

which

have

all

been

related

to

diseased

states.

Other

fluid

mechanic

measures

are

also

discussed

in

order

to

give

further

detail

and

insight

during

postprocessing,

and

may

play

an

important

role

in

the

growth

and

rupture

of

the

aneurysm.

It

is

commonly

accepted

that

certain

fluid

mechanic

measures

in

arterial

haemodynamics

are

linked

to

disease

formation

such

as

atheroma

and

aneurysms,

and

play

a

role

in

vascular

remodelling.

While

the

relationship

between

the

flow

field

and

disease

is

not

fully

understood,

fluid

mechanics

parameters

on

and

near

the

artery

wall,

such

as

wall

shear

stress

(WSS)

and

derived

measures,

residence

times

and

region

of

flow

impingement

are

among

the

most

commonly

sought

correlators

to

disease.

This

is

often

discussed

in

relation

to

mechanotransduction

and

mass

transport.

In

this

work

some

physical

insight

of

these

parameters

is

presented

and

then

related

to

possible

signalling

mechanisms

that

endothelial

cells

may

be

endowed

with.

In

this

work

the

flow

structures

for

two

patient-specific

saccular

cerebral

aneurysm,

shown

in

Figure

are

studied

at

steady-state

and

unsteady

periodic

simulations,

identifying

feasible

means

to

describe

and

extract

relevant

flow

measures..

The

non-planarity

and

tortuousity

of

vessels

play

a

determining

role

in

the

human

arterial

system,

resulting

in

a

strong

influence

of

the

local

vessel

topology

on

the

flow

field

An

abnormal

flow

field,

usually

described

as

complex

and

disturbed,

is

often

related

to

the

diseased

states.

Elevated

WSS

values

have

also

been

associated

with

aneurysm

formation

while

temporal

and

spatial

gradients

and

temporal

directional

fluctuations

of

WSS

have

also

been

studied

in

relation

to

aneurysm

initiation

and

rupture.

Vascular

remodelling

is

thought

partly

as

an

adaptive

response

to

alleviate

undesired

haemodynamic

conditions,

such

as

high

WSS

and

spatial

WSS

gradients

in

aneurysm

formation

or

low

and

disturbed

WSS

in

anastomosis

remodelling

and

atherosclerosis

Flow

structures

and

their

stability

within

the

cardiac

cycle

have

also

been

associated

to

aneurysm

rupture

Despite

the

consensus

that

the

haemodynamic

and

mechanical

properties

in

relation

to

aneurysms

play

a

crucial

role,

there

is

still

substantial

debate

as

to

the

most

appropriate

measures

to

discuss

patient

healthcare

and

risks

of

rupture;

furthermore

the

pathophysiology

of

cerebral

aneurysms

includes

considerations

of

factors

such

as

genetic

predisposition

and

biochemical

reactions

Often

aneurysms

are

studied

from

a

geometrical

standpoint,

since

in

effect

the

shape

of

the

aneurysm

and

the

parent

artery

act

as

boundary

conditions

and

will

determine

the

flow

field

(as

well

as

the

fluid

boundary

conditions

at

artificial

sections).

Among

these

studies

the

curvature

of

the

parent

vessel

upstream

to

the

aneurysm

has

been

correlated

to

aneurysm

rupture

and

often

the

shape

of

the

aneurysm

is

the

most

used

criterion

for

clinical

decisions

In

this

work

methods

based

on

existing

theory

of

critical

points

are

used

to

characterise

concisely

both

the

free-slip

(fluid

away

from

the

wall)

and

no-slip

(fluid

on

or

near

the

wall)

domains.

These

methods

are

shown

to

give

a

description

of

the

flow

field

in

a

simple

approach,

both

to

implement

and

also

as

tools

to

analyse

the

data.

These

methods

are

meant

to

complement

and

assist

the

current

trend

of

studies

of

diseases

of

human

physiology

related

to

haemodynamics..

The

outline

of

the

paper

is

as

follows:

a

brief

outline

of

the

patient

data

sets

and

the

approach

of

reconstructing

a

computational

domain

from

medical

images

is

given

in

Section

The

parameters

for

the

numerical

simulations

are

detailed

in

Section

In

Section

the

outline

of

the

theory

on

the

Taylor

series

expansions

of

the

velocity,

the

wall

shear

stress

and

the

relative

position

is

presented,

and

related

to

critical

point

theory.

The

results

are

presented

in

Section

and

finally

the

conclusions

are

given

in

Section

The

local

dynamics

for

both

the

free-slip

and

near-wall

flow

regions

can

be

studied

using

the

Taylor

series

expansion

of

the

velocity,

wall

shear

stress

or

relative

position.

This

leads

to

a

detailed

local

information

of

the

flow

field

that

can

aid

the

discussion

of

fluid

dynamic

phenomena

concisely.

The

use

of

these

methods

has

been

shown

to

provide

a

clear

and

in-depth

description

of

the

flow

in

two

patient-specific

geometries

of

saccular

cerebral

aneurysms.

The

description

is

similar

to

other

works

involving

vortex

stability

during

the

cardiac

cycle,

wall

shear

stress

magnitude

and

impingement

size

areas,

however

the

approaches

of

extracting

these

results

provided

in

this

work

have

the

advantage

of

being

simple

to

calculate,

provide

a

clear

physical

basis

to

the

measures,

are

robust

and

avoid

a

large

amount

of

clutter

in

the

analysis

that

is

otherwise

inevitable.

The

unsteady

simulations

and

use

of

the

above

described

methods,

have

identified

that

the

WSS

critical

points

(hence

also

the

surface

shear

lines)

alter

noticeably

during

the

cardiac

cycle,

however

the

transport

normal

to

the

wall

and

the

dominant

vortex

core

remain

largely

unaltered.

The

deceleration

phase

of

the

cardiac

cycle

causes

a

larger

number

of

smaller

vortices

to

be

formed

in

the

aneurysm.

The

region

of

flow

separation

at

the

aneurysm

neck

is

not

seen

at

the

peak

systole.

Other

flow

features

that

can

concisely

describe

the

flow

have

also

been

discussed..

An

important

physical

analysis

of

the

WSS

spatial

and

temporal

gradients

from

equation

sheds

light

on

widely

used

fluid

dynamic

parameters

correlated

to

disease.

These

are

related

to

the

near-wall

transport,

and

this

physical

interpretation

can

be

linked

to

possible

sensory

mechanisms

of

the

endothelial

cells,

and

hence

mechanotransduction

and

related

biochemical

signalling..

The

analysis

has

been

performed

on

cerebral

aneurysms

and

can

foreseeably

be

extended

to

study

a

greater

number

of

geometries

and

applications

to

provide

a

more

learned

understanding

of

human

physiology

in

normal

and

diseased

states.

Future

work

would

primarily

be

use

of

a

larger

number

of

data

sets

in

order

extract

correlations

and

statistical

measures

related

to

aneurysm

growth

and

rupture.

Applying

the

above

methods

to

the

steady-state

simulations

for

both

patient

Cases

and

the

time-periodic

simulation

of

patient

Case

the

free-slip

and

no-sip

regions

are

now

analysed.

We

first

present

the

steady-state

solutions

using

the

methods

discussed

above

to

describe

the

key

flow

features

in

the

free-slip

domain

and

importantly

how

these

are

related

to

the

no-slip

domain.

The

results

of

these

steady-state

simulations

are

shown

in

Figure

for

both

Cases

and

those

of

the

unsteady

simulations

are

shown

in

Figure

The

vortex

core

points

appear

unaligned

and

some

single

dispersed

points

are

identified

in

the

flow

field

erroneously;

this

has

been

identified

to

be

due

to

the

interpolation

of

cell-centered

data

obtained

from

the

finite

volume

computations

onto

a

node-centered

data

used

in

the

analysis,

and

from

the

numerical

accuracy

of

the

computational

result

from

which

the

post-processing

of

derivatives

leads

to

additional

errors..

Two

important

points

are

first

addressed

from

the

results

of

Figure

firstly

the

use

of

streamlines

to

describe

the

flow

field,

as

discussed

in

with

relation

to

correlation

to

aneurysm

rupture,

can

be

rather

difficult

to

discern

and

classify

and

a

less

cluttering

approach

is

needed;

secondly

the

λ

criterion

for

vortex

structure

identification

does

not

align

in

fact

to

the

vortex

cores.

This

second

point

can

lead

to

misleading

conclusions

about

the

flow

structures,

the

reason

for

this

is

due

to

the

viscous

terms

in

the

Navier-Stokes

equations,

and

discussed

in

greater

detail

in

such

that

a

pressure

minimum

in

the

plane

(which

is

the

basis

of

the

λ

criterion

may

not

be

appropriate..

The

surface

shear

lines

and

the

critical

points

of

the

WSS,

in

conjunction

to

plots

of

the

WSS

and

TR,

can

be

used

to

describe

the

no-slip

domain

entirely.

The

critical

points

of

the

WSS

act

as

dividing

lines

to

the

surface

shear

lines,

bounding

these

to

common

regions.

The

critical

points

furthermore

identify

locations

of

flow

separation

and

impingement,

and

the

type,

i.e.

if

a

vortex

is

rooted

at

the

wall

or

a

simpler

arrangement

is

present.

The

WSS

and

TR

locally

identify

the

velocity

of

the

fluid

near

the

wall,

in

the

tangential

and

normal

directions.

In

this

way

a

complete

description

of

the

no-slip

domain

is

presented

in

a

concise

and

physically

intuitive

form.

The

vortex

core

points

on

the

other

hand

act

as

skeletonisations

of

the

free-slip

domain

by

identifying

the

axes

of

rotation

(not

always

part

of

a

vortex

therefore),

as

well

as

other

measures

such

as

the

rate

of

stretching,

spiralling

intensity

and

local

osculating

planes,

if

desired.

The

strongest

vortex

cores,

that

leave

the

aneurysm

through

the

side

branch

or

the

main

arterial

vessel,

in

fact

appear

to

be

anchored

to

the

vessel

wall,

as

seen

from

the

critical

points

of

the

WSS.

It

is

further

possible

to

identify

from

the

values

of

TR

and

the

critical

points

of

WSS

the

effect

of

the

swirling

motion

in

the

flow

field

around

the

vortex

cores;

for

example

in

the

case

of

a

vortex

lying

adjacent

to

the

wall

a

net

division

of

sign

of

TR

is

observed,

while

the

surface

shear

lines

would

be

at

an

angle

to

this

division..

In

Figure

snapshots

of

the

solution

at

three

time

intervals

are

presented.

The

interest

is

to

identify

clearly

and

easily

the

characteristics

of

the

flow

field.

An

interesting

result

that

emerges

is

the

relative

constancy

of

TR

during

the

cycle,

while

the

WSS

is

seen

to

vary

more

noticeably.

A

further

distinguishing

feature

is

the

persistence

of

the

vortex

core

anchored

to

the

proximal

portion

of

the

aneurysm

dome

(identified

easily

from

the

WSS

critical

points)

and

continues

to

the

distal

parent

artery.

During

diastole

the

flow

field

in

the

aneurysm

becomes

more

complex

due

to

the

fluid

deceleration,

as

seen

by

an

increased

number

of

WSS

critical

points

and

vortex

cores,

however

by

the

end

of

the

diastolic

phase

(at

the

end

of

the

heartbeat)

the

flow

field

has

again

become

relatively

simplified.

A

more

in-depth

analysis

of

the

time

snapshots

will

be

interesting

to

identify

the

change

in

the

WSS

critical

points

and

the

vortex

cores,

the

behaviour

and

stability

of

which

may

lead

to

correlation

and

classification

of

aneurysm

rupture

or

growth

directions.

B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O


Iterative Schemes for Bump Solutions in a Neural Field Model ABSTRACT. We develop two iteration schemes for construction of localized stationary solutions (bumps) of a one-population Wilson-Cowan model with a smoothed Heaviside firing rate function. The first scheme is based on the fixed point formulation of the stationary Wilson-Cowan model. The second one is formulated in terms of the excitation width of a bump. Using the theory of monotone operators in ordered Banach spaces we justify convergence of both iteration schemes. Neural field models have been the subject of mathematical attention since the publications [1, 2, 3, 4] . These models typically take the form of integro-differential equations. We consider a one-population neural field model of the Wilson-Cowan type [1, 2, 3, 4, 5] (1.1). Here u(x, t) represents the activity of population, f the firing-rate function, ω the connectivity function, and h the firing threshold. For review on the model (1.1) see [5] . Existence and stability of spatially localized solutions and traveling waves are commonly studied for the case when the firing rate function is given by the unit step function [4, 5, 6] . However, the results for the case when the firing rate function is smooth are few and far between [7, 8, 10, 11] .. In the mathematical neuroscience community time-independent spatially localized solutions of (1.1) are referred to as bumps. The motivation for studying bumps stems from the fact that they are believed to be linked to the mechanisms of a short memory [12] . In the case when f is given as a unit step function, one can find analytical expressions for the bump solutions [4] . In principle, bumps solutions can also be constructed when the firing rate function is smooth provided the Fourier-transform of the connectivity function is a real rational function. In that case the model can be converted to a higher order nonlinear differential equation which can be represented as a Hamiltonian system. The bumps are represented then by homoclinic orbits within the framework of these systems. See for example [13, 14, 15] .. Kishimoto and Amari [7] have proved the existence of bump solutions of (1.1) when f is a smooth function of a special type (smoothed Heaviside function), using the Schauder fixed point theorem. The Schauder fixed point theorem, however, does not give a method for construction of the bumps. Pinto and Ermentrout in [9] constructed bumps using singular perturbation analysis. However, this method is quite involved, and is restricted to the lateral-inhibitory connectivity (i.e., ω is assumed to be continuous, integrable and even, with ω(0) > 0 and exactly one positive zero). Coombes and Schmidt in [8] developed an iteration scheme for constructing bumps of the model (1.1) with a smoothed Heaviside function. They, however, did not give a mathematical verification of their approach. Apart from the work of Coombes and Schmidt [8] , the authors of the present paper do not know about other attempts to develop iterative algorithms for the construction of bumps. Thus there is a need for a more rigorous analysis of iteration schemes for bumps. This serves as a motivation for the present work.. We present two different iteration schemes for constructing bumps. The first one is based on the fixed point problem introduced in [7] . The second scheme, which is modification of the procedure introduced in [8] , is an iteration scheme for the excitation width of the bumps. We prove that both schemes converge using the theory of monotone operators in ordered Banach spaces.. The present paper is organized in the following way: In Section 2 the properties of the onepopulation Wilson-Cowan model are reviewed with emphasis on the results of Kishimoto and Amari [7] . In Section 3 some necessary mathematical preliminaries are introduced. Section 4 is devoted to the study of a direct iteration scheme based on the fixed point problem proposed by of Kishimoto and Amari [7] . In Section 4.1 we illustrate the results with a numerical example. In Section 5 we introduce a fixed problem based on the specific representation of the firing rate function studied in [8] . The fixed problem is formulated for the crossing between bumps and a shifted parameterized threshold value h + t, t ≥ 0. The bump solution can be restored from these crossings. We prove that there is a fixed point which can be obtained by iterations.. We provide an numerical example in Section 5.1. In Section 6 we summarize our findings and describe open problems. We have introduced two iteration schemes for finding a bump solution in the f −field of the Wilson-Cowan model: The first scheme is based on the fixed point problem formulated by Kishimoto and Amari [7] . The second one is described by the fixed point problem formulated for the interface dynamics of the bump. The latter formulation became possible due to the special representation of the firing rated function introduced by Coombes and Schmidt [8] .. We have proved using the theory of monotone operators in Banach spaces that both iteration schemes converge under Assumption 1 and 2. From the iterative procedures we obtain the solution on the finite interval [∆ τ , ∆ 0 ] (see Section 4), and on [∆(0), ∆(τ )] (see Section 5). Then it has been shown that under some additional assumptions on the connectivity function ω this solution determines a bump of the f -field on R.. The assumptions imposed for the first method (see Section 4) differ from the ones imposed for the second method (see Section 5) . The evident disadvantage of Assumption 3 and 5 is that they contain information about the output of the iteration procedure, ∆(t). Assumption 3 can be substituted with the more restrictive Assumption 3 , but not Assumption 5 . Thus, Assumption 5 can not be checked in advance. On the other hand, the set of assumptions for the fixed point problem (5.2) is in general less restrictive than the assumptions imposed on the fixed point method outlined in Section 4. All assumptions (except Assumption 5 ) are quite easy to check if ω(x) is given.. We show by a numerical example that both iterative schemes converge to the same solution. Moreover, from numerics it follows that this solution is unique and stable. Indeed, the maximal and minimal fixed points turn out to be equal for any trials and choice of parameters. Thus, by Corollary 3.2, the fixed point is unique. Moreover, the constructed fixed point solution is stable since it is located between stable solutions of the f 0 -and f τ -field, [7] . Notice that we have not given a mathematical verification of these observations.. Notice also that we have looked for the bump solutions under the assumption ∆ τ < ∆ 0 and (2.12). Thus, even if the constructed solution is unique, it does not necessarily mean that there are no other stable or unstable solution. However, the same type of reasoning as we performed here are no longer valid if we relax on these assumptions. Therefore we leave this problem for a future study.. been supported by The Research Council of Norway under the grant No. 178892 (eNEUROmultilevel modeling and simulation of the nervous system) and the grant No. 178901 (Bridging the gap: disclosure, understanding and exploitation of the genotype-phenotype map).

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I I I I O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I I I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Reliable Critical Infrastructure: Multiple Failures for Multicast using Multi-Objective Approach Multicast is the keystone for multimedia Internet. Multicast is one of the new and most used services in telecommunication networks. However, these networks meet big challenges when facing failures from diverse factors, including natural disasters and bad configurations. Networks operators need to establish mechanisms to maintain available multicast services, and plan actions to handle incidents. We study and implement an elitist evolutionary algorithm based on Strength Pareto Evolutionary Algorithm -SPEA. Our implementation recalculates network routes, even when there are multiple failures. The results indicate that our product finds lower-cost and higheravailability multicast tree to protect multicast services. Multicast is one of the new and most used services in telecommunication networks. Some applications on the Internet use a multicast service which is a key factor in multimedia, video conferencing, distributed games, Internet television, and telepresence. A multicast service simultaneously sends messages from only one source to a group of destinations creating a multicast tree over a network. Networks are exposed to several sources of failure, including misconfiguration or operational errors, natural disasters, attacks, and environmental challenges. Additionally, to maintain the services, they must consider unusual but legitimate traffic [1] , such as peak traffic on specific hours or dates. Due to multicast applications demand important amount of traffic, they need a mechanism to protect them from network failures. The Internet, for example, has been designed to survive random failures. If a node is down, the reliable protocol is able to reroute the traffic around and use alternative connections.. In networking, Resilience and Survivability refer to the abilities of the network to overcome failures and maintain its services working. The study of resilience and survivability has become an important aspect of managing infrastructure in multicast networking. In this paper, we formulate a mechanism to protect a multicast service when the network experiences multiple failures. Failures in multicast have been widely studied [2] [3] . A common strategy to face this problem is finding a redundant multicast tree (RMT). Some authors consider that a RMT could be completely link-disjoint from the original. They propose to find a RMT with the minimal cost. To determine the optimal RMT, we consider the NP-hard Steiner tree problem [4] . Several algorithms have been proposed to deal with this issue, such as Topological methods [5] , or the Nearest Participant First (NPF) [6] . Furthermore, there are network protocols which calculate a multicast tree with the minimal cost, such as MOSPF, PIM-DM, PIM-SM, and CBT.. However, there are some limitations with existing approaches. First, the algorithms that calculate RMTs were designed to optimize only one decision variable; thus, they use only distance to find the minimum cost tree. Furthermore, algorithms and protocols previously mentioned were tested with single failures, it means, when only a link disruption occurs. Nonetheless, 80 F.A. Maldonado-Lopez, Y. Donoso telecommunication operators need to find not only routes with the minimal cost, but also routes with high availability. Moreover, operators require techniques to plan and reconfigure the network where a set of links are damaged or eliminated.. We propose a mechanism to protect multicast services, with the maximum availability and the minimum cost, considering multiple failures and diverse decision variables. Our approach applies an heuristic multi-objective evolutionary algorithm (MOEA) which is a stochastic search method to estimate the optimal RMT. We implement the Strength Pareto Evolutionary Algorithm (SPEA) which is an elitist evolutionary algorithm that finds Pareto-optimal solutions [9] . The used method has polynomial computational complexity O(mn 2 ) where m is the number of arcs and n nodes [10] . Result exhibits possible configurations after multiple failure for this problem considered NP-hard.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Artificial Neural Network for Compositional Ionic Liquid Viscosity Prediction Being a new generation of green solvents and high-tech reaction media of the future, ionic liquids have increasingly attracted much attention. Of particular interest in this context are room temperature ionic liquids (in short as ILs in this paper). Due to the relatively high viscosity, ILs is expected to be used in the form of solvent diluted mixture with reduced viscosity in industrial application, where predicting the viscosity of IL mixture has been an important research issue. Different IL mixture and many modelling approaches have been investigated. The objective of this study is to provide an alternative model approach using soft computing technique, i.e., artificial neural network (ANN) model, to predict the compositional viscosity of binary mixtures of ILs [C n -mim][NTf 2 ] with n=4, 6, 8, 10 in methanol and ethanol over the entire range of molar fraction at a broad range of temperatures from T=293.0-328.0K. The results show that the proposed ANN model provides alternative way to predict compositional viscosity successfully with highly improved accuracy and also show its potential to be extensively utilized to predict compositional viscosity taking account of IL alkyl chain length, as well as temperature and compositions simultaneously, i.e., more complex intermolecular interactions between components in which it would be hard or impossible to establish the analytical model. This illustrates the potential application of ANN in the case that the physical and thermodynamic properties are highly non-linear or too complex. Ionic liquids were originally developed with the aim of producing a salt which melts around room temperature, so it is generally defined as a salt consisting only of ions and existing in liquid state under room temperature, 1 such as organic nitrogen-containing heterocylic cations and inorganic anions. 2 Since the first dedicated work on ionic liquids, which was carried out between 1950 and 1990 , an extensive number and range of ionic liquids have been prepared and reported. Numerous subcategories have been defined. The most common subcategory is that of room temperature ionic liquids, salts with melting points of 25 °C or below. Room temperature ionic liquids (RTILs, in short as ILs in this paper), have enormous potential as alternative green solvents for a wide range of applications due to their important and unique properties, such as non-volatile and solvating properties. The rising scientific interest in ILs and their potential applications has triggered an explosive progress in the field of IL synthesis and fundamental research. 3, 4 The rapidly increasing research activity is reflected in the exponentially growing number of publications per year. It is remarkable that papers appear faster than 40 per week underlining the extreme growing interest in this field.. Although a wide range of research regarding ILs has been conducted, there are still a number of challenges on the way to apply ILs in large scale chemical processes. Among them, the separation, transportation and recycling of relatively highly viscous ILs have to be taken into consideration for process development. It should be noted that in most applications, ILs will be used in mixtures with other compounds. For instance, when used as a reaction solvent, the IL will contain both the reactants and the products. The presence of other compounds may significantly reduce the viscosity of the mixture, somewhat ameliorating the disadvantages of the intrinsically higher viscosities of ionic liquids. Consequently, viscosity of IL mixture is important in many future applications and it is imperative to develop knowledge of fluid and transport properties of ILs in mixtures with reactants, products or other co-solvents. Therefore, in addition to the study of fluid and transport properties of pure ionic liquids, studies of viscosity of mixtures of ionic liquids with different class of solvents in binary or tertiary systems are necessary and essential. However, reported studies of rheological behavior of binary mixture of ILs in aqueous, polar or non-polar phase are limited even for some well characterized ionic liquids. 6 In industrial applications, such as in organic synthesis and extraction processes, an efficient ILs mixture viscosity estimation model can help to facilitate processing development prediction with less measurement work. Although various methods are developed for predicting pure compounds viscosities with varying thermodynamic variables, [7] [8] [9] [10] [11] [12] when dealing with mixtures, the modelling to predict viscosity of IL is, by far, a much more complicated problem, 13 and is still on the way of investigation. In most of investigations, predictive compositional viscosity models are proposed to correlate viscosity of the binary mixtures with corresponding viscosity of the pure components. The models are kinds of semi-empirical models to predict the thermodynamic and the physical properties of the compositional viscosity based on theoretical relations and relying on data regression to obtain some empirical parameters, from the experimental data of the binary combinations of the components. The compositional viscosity models have shown worked well from different aspects. Parameters affecting viscosity may vary, many different components (e.g., more than two) can be found in mixtures of ILs and creating a model that accounts for all of them may be unrealistic. For a wide range of temperatures and viscosity compositions, i.e., more complex intermolecular interactions between components, it would be hard to establish analytical model. So in this paper, an ANN is utilized to predict compositional viscosity over a wide range of temperatures and viscosity compositions.. ANNs present significant advantages over conventional data regression models as there is no need to provide a model function, needs no previous knowledge of the data relationship before predicting the desired property. The learning capability of ANNs allows the discovery of more complex and subtle interactions between variables and ANNs are intrinsically more robust when it comes to handling noisy or inaccurate data. [14] [15] Neural networks have particularly proved their ability to solve complex problems with nonlinear relationships. This background makes ANN one of the most perspective techniques of nonlinear data analysis in almost all the fields of chemistry (from quantitative structure-property relationship studies [16] [17] to petroleum chemistry. 18 ANN has been widely applied to predict the physical and thermodynamic properties of chemical compounds. Recently a few researches have been performed by artificial neural networks for prediction of pure substances and petroleum fraction's properties 19 ; activity coefficients of isobaric binary systems 20 ;. thermodynamic properties of refrigerants [21] [22] [23] [24] ; activity coefficient ratio of electrolytes in amino acid's solutions 25 ; the phase stability problem 26 ; and dew point pressure for retrograde gases. 27 Other ANN applications include density predication of ionic liquids 28 ; modeling flow boiling heat transfer of pure fluids 29 ; predicting slag viscosity over a broad range of temperatures and slag compositions 30 ; -T-P prediction for ionic liquids 31 ; prediction of simple physical properties of mixed solvent systems, 32 etc. Those work demonstrated that neural networks can dramatically reduce the numerical errors and eliminate systematic deviation between predicted values and experimental values. Furthermore, industrial application of multivariate data analysis methods is already observed in a number of fields. [33] [34] [35] A complete list of properties has been presented that have been analyzed in the literature using different approaches to artificial neural networks. 36 Properties such as boiling point, critical temperature, critical pressure, vapor pressure, heat capacity, enthalpy of sublimation, heat of vaporization, density, surface tension, viscosity, thermal conductivity, and acentric factor, among others, were thoroughly reviewed. Applications of neural networks to mixture properties (PTV properties, vapor liquid equilibrium, activity coefficients) have been also presented in other publications. 22, 24, [37] [38] There are numerous viscosity measurements available in the literature. Unfortunately, the techniques used to obtain them vary greatly and the sources of potential errors are numerous. To the best of the author knowledge there is not much application for IL viscosity prediction; and on the prediction of these properties for IL's using ANN.. In this research work, the application of ANNs to the prediction of compositional viscosity of IL mixture is investigated. The work has been briefly outlined in Ref. 39 . This paper aims at extending, refining, completing and systematizing the results in Ref. 39 . The paper is organized as follows: ANN is briefly overviewed firstly in Section 2. Some fundamentals and modeling steps of ANN for this application are introduced in Section 3. Materials and experimental approaches are outlined in Section 4. Detailed results and discussion are reported in Section 5. The paper is concluded in Section 6. In this work, the ability of ANN for modeling and prediction of viscosity prediction of ILs mixture has been investigated. The STD analysis based results are used for verification of the suggested approach. Results showed a good agreement between experimental data and the predicted by ANN. An important feature of the model is it doesn't require any theoretical knowledge during the training process. The results show that the presented new neural network models that provide better predictions and higher accuracy than the existing and proposed model. In each case, the ANN does not require a data relationship to predict the compositional viscosity but rather relies on the field data obtained for training. Since the ANN does not rely upon theoretical relations, it can easily be expanded to include other factors such as extra new components.. The work proposed only considered binary mixtures, but since the ANN does not rely upon theoretical relations, it can easily be expanded to the case for ternary and quaternary mixtures and even more new components. In addition, it can also be expanded to include other factors such as IL alkyl chain length, different type of IL or solvent, shear rate for nonNewtonian IL. For example, an attempt has been made to account for ionic liquid alkyl chain length. None of the conventional regression models mentioned previously directly account for both the carbon chain length and measurement condition. With neural networks, the alkyl chain length can be accounted for by one or more additional inputs.. For this reason, although there are some obvious disadvantages of ANN as stated in literatures, it is still recommended that the ANN approach could be applied in ILs fields for reduction in error, computational time, and cost of overproduction and underproduction, because its advantages such as the ability to handle nonlinear data far outweigh the disadvantages.

B I I O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I I O O O O O O O O O O O O B O O O O O O O O O O O O O O O B O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I I O O O O O O O O O B I I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O B O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Hierarchies of Probabilistic and Team FIN-Learning A FIN -learning machine M receives successive values of the function f it is learning and at some moment outputs a conjecture which should be a correct index of f . FIN We show that the problem for given k i ; n i to determine whether k 1 ; n 1 ]FIN k 2 ; n 2 ]FIN is algorithmically solvable. The set of all FIN cut-points is shown to be well-ordered and recursive. Asymmetric teams are introduced and used as both a tool to obtain these results, and are of interest in themselves. To a large extent the study of inductive inference is concerned with with de ning new learning paradigms and comparing their power. Often the only relations which hold between the di erent paradigms of machine learning are those which trivially follow from their de nitions. E.g. in the mindchangeanomaly hierarchy of limit learning EX a b EX c d i a c and b d as shown in CS83] . It was hard to develop a mathematical theory in a situation where almost every time two di erent de nitions lead to two di erent concepts. The initial stage in the research of inductive inference was mostly descriptive: there were more and more new learning paradigms and demonstrations of their di erences.. Probabilistic and team learning changed this situation. In PS88] it was shown that not all probabilistic limit learning types EXhpi are di erent. DKV92a] used \trial and error" to come up with the ratio 24=49, therefore one can ask an interesting (but informal) question: Where does the 24/49 come from? We provide an answer of sorts to this question. This is accomplished by generalizing the problem and casting it in terms of a game. By putting in the corresponding parameters to the game, the constant 24=49 emerges.. Our paper does not focus on nding more constants below 10=21, but rather re ects on the global structure of all cut-points in the interval (0; 1). We describe these cut-points as solutions to combinatorial optimization problems on tree-like objects called widgets. Each widget corresponds to a set of strategies which one team can use to diagonalize against another one. Section 3 focuses on widgets and their use in diagonalization and simulation. Section 4 shows that asymmetric teams are of independent interest, since \natural" questions about symmetric teams may yield asymmetric teams as intermediate results. In this work we reduce the problem of comparing FIN teams to combinatorial optimization on unusual objects we call widgets. Cut-points for probabilistic FIN-learning are then related to the partial order of widgets. The whole work turns out to be a survey of combinatorial methods used in team inductive inference.. Extending the traditional notion of symmetric teams to arbitrary (symmetric or asymmetric) teams described by matrices illustrates the idea advocated by Polya ?] that a generalized statement can be easier and more fun to prove than a restricted one. Polya calls this \balancing the inductive step". To nd when the inclusion k 1 ; n 1 ]FIN k 2 ; n 2 ]FIN takes place, it can be better to solve a more general problem about asymmetric teams (see Theorems 33 and 40).. Moreover, asymmetric teams can describe many team-related concepts of learning which so far lacked a common methodology. In arithmetic complex numbers are introduced, because they are closed with respect to a larger set of arithmetic operations, and many problems formulated in terms of real numbers can be easily solved in the complex space and interpreting the results appropriately.. There are hardly any operations we can do with symmetric team types k; n]I. ( DK96] considers redundi cation where k; n]FIN is replaced by mk; mn]FIN where m is positive integer.) Asymmetric team types are closed under intersection and memberwise intersection (Theorems 56 and 50), memberwise union (Theorem 51). They are not closed under union. That can be due to some fundamental reason, in analogy with convex sets or linear subspaces not being closed against union.. Open problems in this area of inductive inference include: generalization of diagonalization and simulation results, e.g. to include EX n for any n, generalization of team matrices allowing matrix entries to be any real numbers in 0; 1] rather than just 0 and 1, thus merging team and probabilistic learning.. Complete picture of probabilistic FIN hierarchy, nding the probability up to where H FIN is well-ordered.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O


An Extended Path Following Algorithm for Graph-Matching Problem Abstract-The path following algorithm was proposed recently to approximately solve the matching problems on undirected graph models and exhibited a state-ofthe-art performance on matching accuracy. In this paper, we extend the path following algorithm to the matching problems on directed graph models by proposing a concave relaxation for the problem. Based on the concave and convex relaxations, a series of objective functions are constructed, and the FrankWolfe algorithm is then utilized to minimize them. Several experiments on synthetic and real data witness the validity of the extended path following algorithm. GRAPH matching plays a central role in many graph-based techniques, such as classification in structural pattern recognition, where patterns are represented by graph models [1] , and identification of pairs of homologous proteins, where proteins of each species are organized by a graph [2] . Given two graphs G D ¼ ðV D ; E D Þ and G M ¼ ðV M ; E M Þ, where V and E, respectively, denote the sets of vertices and edges, matching between them is to find a one-to-one mapping f : V D ! V M such that some fitness function is minimized. The fitness function can be defined in different ways, such as on the edit distance [3] and Frobenius norm of the difference between adjacency matrices [4] . The problem is, in nature, an NP-hard combinatorial optimization problem with factorial complexity, except for some graph models with special structure, of which a typical example is the planar graphs, which have been shown to be of polynomial complexity [5] .. To make the problem computationally tractable, many approximate approaches have been proposed in the last three decades, trying to seek an acceptable tradeoff between the complexity and matching accuracy. The approximate approaches can be categorized from different perspectives, for which an extensive review is referred to [6] , for instance. In this paper, we focus on one group of approximate techniques that works directly on the adjacency matrix of the graphs to be matched and involves a relaxation of the combinatory optimization to be a continuous one [7] , [8] , [4] . With the help of the adjacency matrix, the matching problem can be formulated as identifying a permutation matrix that minimizes some cost function as follows:. where P denotes the set of permutation matrices, A the adjacency matrix, and A P ðMÞ the permuted adjacency matrix of A M by P . To avoid the combinatory nature of the problem, one typical approach is to relax the domain of the problem from P to its convex hull, i.e., the set of doubly stochastic matrices denoted by D. However, optimization of the relaxed problem generally results in a doubly stochastic matrix instead of a permutation one. To finally solve the matching problem, a further step is usually needed to project the doubly stochastic matrix P back to a permutation matrix X. An intuitive approach may resort to solving the following maximal linear assignment problem:. typically with the help of the Hungarian algorithm [9] . Such a hard-cut type of assignment may, however, introduce a significant additional error. By contrast, starting from some appropriate P 2 D, the graduated assignment [8] and PATH following [4] algorithms can converge to a permutation matrix in a smooth way. The graduated assignment algorithm approaches a permutation matrix by increasing a parameter which controls the nonconvexity of the problem, and as the parameter is increased to be large enough, the algorithm finally gets a permutation matrix. On the other hand, a concave relaxation of the original graph-matching problem was introduced by the PATH following algorithm, and by linearly combining the concave and convex relaxations, the PATH following algorithm provides a gradual nonconvexity-like algorithm to minimize the concave relaxation which holds the same minima as the original matching problem. For matching problems on undirected graphs, the PATH following algorithm exhibited an excellent performance on matching accuracy. However, the PATH following algorithm cannot be used to solve the matching problem on directed graph models since the concave relaxation proposed in [4] can no longer be guaranteed to be concave. In this paper, we propose a concave relaxation for the directed graph model, and thus extend the PATH following algorithm to cover such a type of graphs. A brief review of the PATH following algorithm is given in Section 2. The proposed method is described in detail in Section 3, followed by some experimental demonstrations given in Section 4. Section 5 finally concludes the paper. The PATH following algorithm was recently proposed to solve the matching problem on undirected graphs and exhibited excellent performance on matching accuracy. In this paper, we extend the algorithm to cover the directed graph models by proposing a concave relaxation for the directed graphs without self-loops. Some experimental comparisons on synthetic and real data witness the validity the proposed method.

O O B I I O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O B I O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O B I I O O O O O O O O O O B I I O O O O O O O O B I O O O O O O O O O O O O O O O O O B I I O O O O O O O O O B I I O O O O O O O O O B I O O O B I O O O O O O O O O O O O O O O O O B I O O B I O O O O O B I I O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O B I O O O O O O O O O O O O O O O O O O O O