text:
Detroit Tigers notes: JaCoby Jones to start in CF vs. LHPs, some RHPs
Anthony Fenech
JUPITER, Fla. – When the Detroit Tigers arrived in Florida some seven weeks ago with any number of options to fill their starting centerfield spot, rookie JaCoby Jones didn't seem like the most likely candidate.
At 23, Jones is still young. He has only 28 big league at-bats and the front office would have preferred he develop a bit longer in the minor leagues.
But as the Grapefruit League season rolled on, Jones – with his awesome athletic ability and natural instincts in centerfield – picked up momentum in the race. None of the other competitors – notably Tyler Collins or Mikie Mahtook – gained ground.
And after J.D. Martinez went down with a right foot injury two weeks ago, Jones was thrust into the driver's seat.
On Friday afternoon, manager Brad Ausmus announced Jones made the team.
"I think certainly the injury to J.D. changed our plans a little bit," Ausmus said. "In terms of defense (Jones) would be a huge asset at Comerica (Park). He swung the bat pretty well, so I think that boils down to the reason."
JaCoby Jones' energy adds spark to Detroit Tigers' veteran clubhouse
Jones will share time with Andrew Romine in centerfield, playing against left-handed pitchers and "certain" righties, Ausmus said.
The decision means the left-handed-hitting Collins and righty Mahtook will platoon in rightfield while Martinez is out, likely for the first month of the season.
In 23 spring games entering Friday, Jones was hitting .327 with two home runs, a double and a triple. He has three stolen bases.
"He's still learning himself," Ausmus said. "He certainly hasn't learned the league yet. There's going to be some bumps, there's going to be some days with his athleticism and just because of his extreme athletic ability that he's going to look extremely talented. So it will be a bit of a roller coaster probably, but that's fine. That's part of learning the game at the major league level as a young player."
Ausmus said Collins and Mahtook will be limited to rightfield for the time being, unless a need emerges in centerfield.
Jones was acquired at the trade deadline in 2015 from the Pittsburgh Pirates for right-handed reliever Joakim Soria. In the nearly two seasons that followed, he has risen through the organization's ranks. He played in 13 major league games last season.
The Tigers intend to be careful with Jones, knowing how tough the everyday transition to the major leagues is. Last September, he went 6-for-28 with no walks and 12 strikeouts.
"We're trying to win," Ausmus said. "So we're going to put the lineup out there and defense out there, some combination of those two factors, that help us win.
"We certainly don't want to stunt his development. Hopefully, we can put him in a position to succeed at times against the pitchers he plays against and he can track some balls in centerfield to help our pitchers out."
Clearing waivers: Both outfielder Steven Moya and left-handed reliever Daniel Stumpf cleared waivers Friday afternoon. Moya was outrighted to Triple-A Toledo. Stumpf was offered back to the Kansas City Royals but will elect free agency and could return to Detroit.
"I'm glad we have him, but you never know what's going to happen," Ausmus said.
He spoke with Moya before the team hopped on the bus for the final two games of the spring.
"I said, 'Personally, I hope I see you again soon, but if I don't, that means you're in the big leagues,' " Ausmus said.
Still mum: Ausmus still wasn't ready to announce the starting pitchers for the second and third games of the season against the Chicago White Sox. They are expected to be right-hander Jordan Zimmermann and left-hander Daniel Norris. Justin Verlander will start Monday's opener.
"Waiting on some things," Ausmus said.
Matthew Boyd's arsenal keeps hitters guessing
Here's why Detroit Tigers' Miguel Cabrera might actually be underrated
Contact Anthony Fenech: afenech@freepress.com. Follow him on Twitter @anthonyfenech.


text:
Why study psychology?
Honours in Psychology
Master of Arts in Psychology (Thesis)
Master of Arts in Clinical Psychology
Master of Philosophy in Public Mental Health
PhD in Psychology
2017 Research Report
Master of Arts in Clinical Psychologyhagen2022-05-20T09:52:20+02:00
Programme Information
Course Coordinator: Professor Elmien Lesch
Selection Coordinator: Ms Megan Snow – mclinical@sun.ac.za; 021 808 3461/ 021 808 2696
Postgraduate Administrative Officer: Ms. Gaynohl Andrews – andrews@sun.ac.za; 021 808 34641
Application Process and Admissions
Prospective students have to meet the following criteria in order to apply for the Masters programme in Clinical Psychology:
An Honours degree in Psychology or an equivalent that Senate has approved for admission to the programme with a minimum average of 65%
Proficiency in at least two official languages
Specific application processes and closing dates are to be updated on the website closer to the time. However, applications generally include the following:
Online application form (please read through the application instructions below, the link is available in the document)
Two independent referee reports
Certified academic records
Black and white photograph of yourself
Application Process and Admissions for 2022
Online Applications for the MA (Masters in Clinical Psychology)are now open. Read the following documents carefully before proceeding with your application:
APPLICATION INSTRUCTIONS 2022 (.docx | 42.6 Kb)
Application-Form – Stellenbosch 20220520 (.doc | 157 Kb)
Open as read only, save to your computer and complete.
REFEREE REPORT 2022 (.docx | 85.2 Kb)
The closing date for applications is Monday 30 May 2022.
No late or incomplete applications will be considered.
The selection week will run from 15 August – 17 August 2022.
An Honours degree in Psychology or an equivalent qualification that Senate has approved for admission to the programme, or a Master's degree in Psychology, completed with a minimum final mark of 65% (current Honours students may also apply on condition that they successfully complete the degree at the end of 2022)
Proficiency in at least two official languages.
Please note that we are unable to answer any telephonic queries at this point. Please forward any queries to our email (mclinical@sun.ac.za) and we will respond within 2-3 working days.
Feedback on pre-selection will be communicated via email. The date of this feedback will be communicated on our website.
If you consider to sign up for this programme please download and read the following document with further information:
MA Clinical Psychology – Suggestions to prospective applicants (pdf | 67 Kb)
The programme focuses on the identification and treatment of psychopathology in children and adults, as well as on the development, implementation and assessment of preventative and remedial programmes. The theoretical component is presented in English and consists of different modules which cover the broad terrain of Clinical Psychology. The practical component of the programme requires you to be involved in the diagnostic assessment and psychotherapeutic treatment of adults, children and families, and to undertake individual community projects. You must be involved continuously throughout the year (approximately 15 to 20 hours per week) with the Welgevallen Community Psychological Clinic and within departmental community projects. A research assignment, based on independent research under supervision, must also be completed.
This programme satisfies the conditions and requirements of the Professional Board for Psychology for the professional training of clinical psychologists and is accredited as such with the Professional Board. In order to register as a clinical or psychologist the Professional Board for Psychology requires:
The successful completion of examinations and practical work;
A full-time internship of 12 months accredited by the Professional Board;
12 months of community service; and
The successful completion of the Professional Board's examinations for psychologists.
Registration with the Board as a clinical psychologist is determined by whether an accredited clinical internship has been completed. Details about internships and community service is available on the Health Professions Council of South Africa (HPCSA): Professional Board for Psychology website:
https://www.hpcsa.co.za/?contentId=0&menuSubId=52&actionName=Professional%20Boards
Programme Duration
The programme is presented full-time for the duration of one academic year.
Compulsory Modules and Credits
Research Assignment (60 credits)
Professional Development (30 credits)
Psychological Assessment (50 credits)
Psychological Interventions (40 credits)
Assessment and Examination
A minimum final mark of 50% is required to pass each module and the final oral examination. The final mark for the programme is based on the relative weights of the different modules as indicated by their respective credit values. In addition, the Professional Board for Psychology requires that, upon completion of the programme, your knowledge and skills be externally assessed and marks be moderated. Practical work is assessed continuously, and all practical work must be successfully completed as a prerequisite for obtaining the degree.
© Copyright 1985 - by Stellenbosch University | All Rights Reserved
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redpajama_set_name: RedPajamaCommonCrawl


text:
Q: JQuery create new select option I have the below functions in regular JavaScript creating select options. Is there a way I can do this with jQuery without having to use the form object? Perhaps storing the options as an array of JSON objects and parsing this in the calling function...
function populate(form)
{
form.options.length = 0;
form.options[0] = new Option("Select a city / town in Sweden","");
form.options[1] = new Option("Melbourne","Melbourne");
}

Below is how I call the function above:
populate(document.form.county); //county is the id of the dropdownlist to populate.

A: Something like:
function populate(selector) {
  $(selector)
    .append('<option value="foo">foo</option>')
    .append('<option value="bar">bar</option>')
}

populate('#myform .myselect');

Or even:
$.fn.populate = function() {
  $(this)
    .append('<option value="foo">foo</option>')
    .append('<option value="bar">bar</option>')
}

$('#myform .myselect').populate();

A: A really simple way to do this...
// create the option
var opt = $("<option>").val("myvalue").text("my text");

//append option to the select element
$(#my-select).append(opt);

This could be done in lots of ways, even in a single line if really you want to.

A: What about
var option = $('<option/>');
option.attr({ 'value': 'myValue' }).text('myText');
$('#county').append(option);

A: This is confusing. When you say "form object", do you mean "<select> element"? If not, your code won't work, so I'll assume your form variable is in fact a reference to a <select> element. Why do you want to rewrite this code? What you have has worked in all scriptable browsers since around 1996, and won't stop working any time soon. Doing it with jQuery will immediately make your code slower, more error-prone and less compatible across browsers.
Here's a function that uses your current code as a starting point and populates a <select> element from an object:
<select id="mySelect"></select>

<script type="text/javascript>

function populateSelect(select, optionsData) {
    var options = select.options, o, selected;
    options.length = 0;
    for (var i = 0, len = optionsData.length; i < len; ++i) {
        o = optionsData[i];
        selected = !!o.selected;
        options[i] = new Option(o.text, o.value, selected, selected);
    }
}

var optionsData = [
    {
        text: "Select a city / town in Sweden",
        value: ""
    },
    {
        text: "Melbourne",
        value: "Melbourne",
        selected: true
    }
];

populateSelect(document.getElementById("mySelect"), optionsData);

</script>

A: How about
$('#county').append(
    $('<option />')
        .text('Select a city / town in Sweden')
        .val(''),
    $('<option />')
        .text('Melbourne')
        .val('Melbourne')
);

A: If you need to make single element you can use this construction:
$('<option/>', {
    'class': this.dataID,
    'text': this.s_dataValue
}).appendTo('.subCategory');

But if you need to print many elements you can use this construction:
function printOptions(arr){
    jQuery.each(arr, function(){
        $('<option/>', {
            'value': this.dataID,
            'text': this.s_dataValue
        }).appendTo('.subCategory');
    });
}

redpajama_set_name: RedPajamaStackExchange


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text:
Last seen: 約1ヶ月 前 | 2021 以来アクティブ
11 質問
Awarded to Jani on 22 Aug 2021
MATLAB Answers (11)
I want to make a GUI in which only those functions run whose radio button is on, but how to make it?
Suppose I have 5 differential equations, but I only want to run and plot those, whose radio button is set 'on', and it's importa...
約2ヶ月 前 | 1 件の回答 | 0
GUI - ODE problem, can someone please help me solve an ode system using guide?
I want to make a user interface with gui, which solves a sysmet of differential equations, and some of the variables (f1 and f2)...
How to discretize a system of spatial and temporal differential equation for mass and heat transfer whit the Euler method?
I want to make a model for the equations which are found in the attached article, and have some questions regarding them. Here ...
2ヶ月 前 | 0 件の回答 | 0
How to solve a differential equation system with time and space derivates? Where to write (i, j) (i-1, j) (i, j-1)?
Hi! I am modelling the drying of granulates/particles based on the attached article and equation system which contains time and...
How to plot a circle with polar coordinates in which the colour of the circle changes by data?
Hi! I want to make something like this, which I found in an article. In my work I model the drying of granulates, and the moi...
How can I make a circle which changes its colour with its properties? I show in the attached figure what I mean.
So I modelled how the mositure changes inside a spherical particle as a function of the particle's radius. I have a lot of da...
How can you solve a system of second order differential equation, which is not time, but length and radius dependent?
I thnik it cannot be solved by some prebuild ode integrator, like ode45. I tried using 'for loop', but I don't know, how to solv...
Solving a system of second order differential equations, with length and radial derivates, but the conditions are quite complex to me.
Hi! I want to use for loop to solve these equations (for drying particles in a fluid bed), but there are two things which preve...
Solving system of non-linear partial differential and algebraic equations.
Hi guys! I already had a couple of questions about my fluid-bed drying topic. It is almost over, and I understand pretty much a...
How can a differential equation be solved, if its constants are in a system of equation relation?
So the whole equations are here, but for the sake of simplicity I reduced many of the constants, so I can focus only on the rele...
Why I can't integrate a matrix in Matlab?
So for a while it worked, but than I ran the program again, and somehow it doesn't work again, though I did not change the code....


Linoleum is the only floor covering marketplace, consisting mainly of organic materials. You can even select how thick you want the Linoleum to be. Also the designs that it comes in are wonderful with marble, wood and tile effects all readily offered. So if you have not believed about making use of Linoleum and you are seeking for new flooring tips, then now might be the time to reconsider your selections. This tends to make it a nice choice for places far more prone to bacteria such as the kitchen and bathroom.
Most people who select vinyl sheet or luxury vinyl more than linoleum do so because vinyl is less complicated to retain. Qualified Installation Is Advised – specialist installers realize how to operate with the material, such as minimizing and sealing the seams so that you get the ideal installation achievable. When vinyl sheet flooring came on the scene, a lot of individuals believed vinyl was the new" linoleum, but they are really unique goods. Other properties: One particular of the most distinguishing features of vinyl when compared to linoleum is that vinyl flooring is totally waterproof.
Linoleum utilised to be the selection only in commercial spaces like hospitals and schools, but is now experiencing a resurgence as a green option to vinyl. Carpet used to be a frequent kitchen flooring option, but the capacity to keep it clean from spills quickly proved to be a mistake. The longevity of vinyl flooring is largely dependent on its protective coating.
For this explanation, linoleum may well be preferable to shoppers who are environmentally conscious. Tiles use the Uniclic joint which is used in other sorts of floating floors like laminate, wood and cork. Vinyl flooring installed prior to the late 1980s may possibly include asbestos and call for a specialized approach for safe removal. Today's linoleum floors can be fashionable or understated, based on your style and tastes.
Vinyl is created from petrochemicals where linoleum is made from linseed oil combined with wood flour or cork dust, ground up limstone and colour pigments on a canvas or jute backing. The advantages of installing a laminate floor as opposed to a hardwood floor in the kitchen is that the laminate floor is much more durable and water resistant than the hardwood floor would be. Upkeep: Linoleum flooring needs slightly far more upkeep than vinyl floor, but cleaning is nonetheless reasonably effortless. Vinyl wears nicely, resists water and is 1 of the most economical flooring alternatives offered.


is slow to respond. is disorganized. is too expensive. are poor communicators. is me, or another co-worker. is fine, but I want the free IT consult.
SCHEDULE YOUR FREE IT CONSULTATION VALUED AT $500 FOR A LIMITED TIME!


text:
Home Obituaries Obituary: Bob Beebe
Obituary: Bob Beebe
It is with deep sorrow that the family of Bob Beebe, a 30-year resident of Idyllwild, relates the news of his passing Tuesday, March 3, 2009, at his home in Hemet.
Bob came to Idyllwild as a teacher in language and music at the famed Desert Sun School, but very soon became recognized as a gifted teacher in the ways of kindness and responsibility for all who came in contact with him.
Though Bob did much in the way of rebuilding and reselling crippled properties on the Hill, his real passions were classical music at his piano, producing his unique works of painting as a self-taught artist, his love of old cars, and his deep reading of all kinds of books that might offer insight into "the human condition" and ways of improving it. Bob was a seeker, a learner and rich in empathy and compassion; he was a great teacher by example.
But by far, Bob's greatest passion and satisfaction was his family — his wife WenPing (Mary) from China, whom he met on the Hill, and their two children, Melissa, 9, and Mark, 7.
Some of the knots in his early rope are these: Born in Coos Bay, Ore., in 1931, he and his brothers were raised through the Great Depression by a strong-willed aunt. After service in the U.S. Army in Germany, he attended Humboldt State College and Mexico City College, and left the University of Colorado with a master's degree. He taught at Deep Springs College before coming to the Desert Sun School in Idyllwild.
For all those wishing to attend, there will be a wake in remembrance of Bob at his home. If you would like to attend, please contact Mary Beebe at (951) 927-7325 for the date, time and directions.
Previous articleObituary: Bonnar Blong
Next articleObituary: Mary Austin


Nashville documentarian and music writer Craig Havighurst says that Annie Crane is carrying on the tradition of "refashioning folk music in a modernist and urbane guise". Compared to the likes of Sandy Denny, Lucinda Williams and Gillian Welch, Crane writes of immigrant grandparents, heartache in snowy Toronto and of an industrially strong Brooklyn love. She has played the Mercury Lounge, Rockwood Musichall and countless other famed NYC venues, has been featured at the New Jersey Folk Festival two years in a row, closed out the 2009 Rochester Indie Festival, toured internationally, performed on numerous radio programs (Americana Crossroads in KY, the Blue Plate Special in TN, Over Yonder in MA) and shared the stage with The Bowmans, Phoebe Krutz, Au Revoir Simone and most recently, with Emmylou Harris at the inaugural "Music City Roots: Live from the Loveless Cafe" in Nashville TN.
"Through the Farmlands & the Cities", Crane's debut full-length, full band endeavor, was recorded within the urban cabin of GaluminumFoil Studios in Brooklyn, NY from November 2008-February 2009. Co-produced with Chris Cubeta, the CD holds true to Crane's folk/Americana roots while incorporating a band that lends profundity without pulling the listener away from the lyrics.


text:
Campaign Launched to Save Our Bingo Clubs
Home » Campaign Launched to Save Our Bingo Clubs
The bingo association has launched a campaign to keep bingo clubs open. As the Scottish government new five-tier alert system of coronavirus restrictions is expected to be implemented today (Tuesday), UK bingo clubs and the future of bingo are under real threat.
A website – https://bingoismylifeline.co.uk/scotland/ – has been set up by the bingo association, with the aim of lobbying Nicola Sturgeon and the Scottish government to keep the bingo halls open under the new lockdown tier system.
Titled 'Bingo is my lifeline – 48 hours to save Bingo in Scotland', the campaign makes the case:
"With loneliness on the rise, many people need COVID-19 safe spaces to overcome social isolation and connect with others. Indeed, 1 in 10 bingo club members say they can't imagine life without bingo and 3 in 10 say it is the highlight of their week.
"If you don't want your local bingo club to close under Level 2 restrictions, please help us by making the case to the Scottish Government that bingo is safe to remain open and is a lifeline for yourself and your fellow bingo players."
The website helps you contact Nicola Sturgeon explaining why bingo halls are essential to the community. There is also a national campaign where you can write to Robert Jenrick MP, the Secretary of State for Housing, Communities and Local Government to protest the contradictory measures which see Liverpool bingo clubs open, but dozens of clubs in the North West close.
"Open This Evening, Closed Again Next Week
Ryann Stisi, owner of the Border Bingo Club in Galashiels, Scotland said:
We're very excited about opening tonight as we've been closed for seven months.
"We've had to cut our capacity from 90 (seats) to 30 to comply with social-distancing, so it's been very difficult. But one of the main reasons for reopening now is to ensure none of our staff are made redundant.
"We also have a duty to our customers; living in a rural area can be quite isolating – there's not a lot to do in the evenings – and this is a chance for them to have a bit of fun and see their friends again."
"We find ourself in a position where we open up this evening and, with this new tiered system, we could be closed again next week."
For Ryann and other bingo fans and employees, a lockdown is bitter pill to swallow. The inconsistency of the restrictions would see Border Bingo Club close should the region be placed into Level Two, Meanwhile, the cinema next door, housed in the very same building, would be permitted to stay open.
With the retail bingo industry reeling from lockdowns and fewer visitors, it is time to join the campaign to save our bingo clubs. For information and to join the campaign, visit here.


A Junior Chess event is to be held at the Uckfield Youth Centre, next to Uckfield Community Technology College (UCTC), Downsview Crescent, Uckfield TN22 3DJ.
The event will be a training day entitled `Rook Raider' and mainly focused on improving skills in the use of the rook in chess, so will be of benefit to players of all abilities including beginners, but will involve learning through competitive play and it will incorporate our ever popular annual Easter Egg fun competition. A Top School trophy will be awarded at the final presentation, but there will also be individual prizes for year groups as well as the top three in the main competition element, plus of course eggstasy for some. This event is suitable to all who can play the game and is open to all junior ages and abilities.
The object of the event is to encourage all levels and ages of junior players to play in and to increase their experience and enjoyment of tournament Chess and meet others of like mind in this area.
The cost for the event will be £10. Cheques should be made payable to Sussex Junior Chess.
Sussex Junior Chess are not responsible for transport to/from the Centre and children attending will need a packed lunch and drinks for the day. Refreshments will not be supplied.
Disclaimer: Organisers and helpers accept no responsibility for any damage, theft or accident to the person or property of any competitor or visitor at this event. Nor are they responsible for the effects of actions of anyone else attending this event.
Supervision: Organisers and helpers cannot act 'in loco parentis' for juniors. Parents/guardians are responsible for their children at all times and in all places during this event.
The deadline for entries is the first post on Saturday 6th April 2019.
Mr Brian Stockham, 36 Scarletts Close, Rocks Park, Uckfield, East Sussex TN22 2BA.


The leader of a luxury car theft gang and his accomplice were shot dead by the police near Fairview Apartments, USJ 22, yesterday. 35-year-old Rizlan Ahmad Tajuddin, also known as Jack, and 34-year-old Mohamed Fadzil Mohamed were stopped by police in a stolen Toyota Vios vehicle. The police were acting on a tip-off that the suspects were on the way to steal another car.
Immediately, the men fired two shots and drove off. When their car was cornered along Jalan USJ 22/1, police fired shots at them as they tried to force their way out. Fadzil, who has 12 criminal charges, died at the scene. Rizlan died on the way to the hospital. He had 3 previous convictions.
Police found a .45 semi-automatic pistol, three .45 caliber rounds, 7 bullets and some fake registration plates in the stolen car.
The armed robbery gang, commonly known as "Gang Jack", are believed to be involved in at least 40 carjacking cases. The gang had 11 members. One of them was shot dead in Kampung Subang, while another 3 were arrested.
It is believed that the gang would sell the stolen cars for RM25,000 each on the black market.


Have you heard that white pupils in photos of children have resulted in discovery of cancerous eye tumors which many times mean removal of the child's eye?
I've been hearing a lot about that on the news lately. Any personal experience or thoughts on this?
Haven't heard anything about it, but it's probably just another scare tactic to get more money. Makes me sick to think how many healthy eyes got gouged out of a helpless child to stuff Big Pharma's pockets.


text:
\section*{Introduction}
Let $X$ be a complex manifold. We denote by $\mathcal{O}_X$ (resp. $\mathcal{D}_X$) the sheaf of holomorphic functions (resp. holomorphic differential operators) on $X$. For any reduced divisor $Z$ in $X$, we denote by $\mathcal{O}_X[\ast Z]$ the sheaf of meromorphic functions with poles along $Z$ at most. Given a regular holonomic $\mathcal{D}_X$-module $\mathcal{M}$ and a meromorphic function $g$ on $X$ with poles along $Z$, we define the $\mathcal{D}_X$-module $\mathcal{M}e^g$ as being the $\mathcal{O}_X$-module $\mathcal{M}[\ast Z]:=\mathcal{O}_X[\ast Z]\otimes_{\mathcal{O}_X}\mathcal{M}$ with connection $\nabla_g$ defined as $\nabla+dg$, where $\nabla$ is the connection on $\mathcal{M}[\ast Z]$ given by its left $\mathcal{D}_X$-module structure. It is known that $\mathcal{M}e^g$ is a holonomic $\mathcal{D}_X$-module. It is irregular along $Z$. We will say $\mathcal{M}e^g$ has \emph{exponential type}.

In the algebraic setting, if $U$ is a smooth algebraic variety over $\mathbb{C}$, we use an exponent ``$\mathrm{alg}$'' in the notation of the corresponding sheaves and we define the notion of a holonomic $\mathcal{D}_X^{\mathrm{alg}}$-module of exponential type similarly.

Let $\mathcal{M}e^g$ be a holonomic $\mathcal{D}_X$-module of exponential type. The cohomology sheaves of its direct image by any proper map $f:X\to C$ to a complex curve are holonomic $\mathcal{D}_C$-modules with possibly irregular singularities. The main result of this article consists in the computation of the formal invariants of these cohomology modules at their singularities.

The basic computation takes place in the following setting: the complex manifold $X$ is the product $D\times \mathbb{P}^1$ of a small disc $D$ centered at the origin in $\mathbb{C}$ by the Riemann sphere, the meromorphic function $g$ is the projection $p_2:X\to\mathbb{P}^1$ with polar divisor $Z=D\times\{\infty\}$ and the map $f$ is the projection $p_1:X\to D$. We will denote by $t$ a coordinate on $D$. As above, $\mathcal{M}$ is a regular holonomic $\mathcal{D}_X$-module. The only interesting  cohomology module to consider is $\mathcal N=\mathcal{H}^0p_{1+}(\mathcal{M}e^{p_2})$ and we assume that $D$ is small enough so that $0$ is the only singular point of $\mathcal{N}$.

In a neighbourhood $U$ of $(0,\infty)\in D\times\mathbb{P}^1$, the singular support of $\mathcal{M}$ is an analytic curve $S$. We denote by $\{S_\ell\mid\ell\in\Lambda\}$ the set consisting of the local irreducible components of $S$ which are distinct of $\{0\}\times\mathbb{P}^1$ or $D\times\{\infty\}$. Set $U^{\ast}=U\setminus S$. To any $S_\ell$ at $(0,\infty)$ we associate:
\begin{figure}[h]
\includegraphics[height=2.4cm]{Zlbis.eps}
\end{figure}
\begin{itemize}
\item
$m_\ell\in\mathbb{N}^*$:  the multiplicity of the conormal space $T^*_{S_\ell}X$ in the characteristic cycle of $\mathcal{M}$.
\item
$p_\ell,q_\ell\in\mathbb{N}^*$: the intersection multiplicity at $(0,\infty)$ of $S_\ell$ with $\{0\}\times \mathbb{P}^1$ and $D\times \{\infty\}$, respectively.
\item
$Q_\ell\subset\mathbb{R}^2$: the convex hull in $\mathbb{R}^2$ of the union of $Q:=\{(u,v)\in\mathbb{R}^2\mid u\leq0,\,v\geq0\}$ and $(m_\ell p_\ell,m_\ell q_\ell)+Q$.
\item $\alpha_\ell\in\tau^{-1}\mathbb{C}[\tau^{-1}]$ and $\delta_\ell\in\mathbb{C}\{\tau\}$: the polar part and the holomorphic part of a Puiseux parametrization of $S_\ell$ at $(0,\infty)$. Let $u_{\ell}\in\mathbb{C}\{\tau\}$, $u_\ell(0)\neq 0$, such that $\gamma(\tau):=(\tau^{p_\ell},\tau^{-q_\ell}u_\ell)$ is a parametrization of $S_\ell$. $\alpha_\ell$ is the polar part of $\tau^{-q_\ell}u_\ell$ and $\delta_\ell$ is its holomorphic part ($\tau^{-q_\ell}u_\ell=\alpha_\ell+\delta_\ell$).
\end{itemize}

The irregularity number of $\mathcal N_0$ is computed in [9]. But, we can go further in computing formal invariants of $\mathcal N_0$.

\begin{theorem}\label{1}\mbox{}\par
$(1)$ After a suitable translation, the Newton polygon of the $\mathcal D_{D,0}$-module $\mathcal N_0$ is the Minkowski sum $\sum_{\ell\in\Lambda}Q_\ell$. In particular, the set of slopes of $\mathcal N_0$ is $\{q_\ell/p_\ell\mid\ell\in\Lambda\}$ and the irregularity number of $\mathcal N_0$ is $\sum_{\ell\in\Lambda}m_\ell q_\ell$.

$(2)$ Let $p=\mathrm{lcm}\{p_\ell\mid\ell\in\Lambda\}$. After the ramification $\rho:D^{'}\to D$, $\rho(\tau)=t=\tau^p$, the formal irregular part of $\rho^*\mathcal N_0$ decomposes as $\oplus_{\alpha\in\Gamma}R_\alpha e^\alpha$ where:
\begin{itemize}
\item
$\Gamma\subset\tau^{-1}\mathbb C[\tau^{-1}]$ is a finite subset and $\alpha\in\Gamma$ if and only if there exist $\ell\in\Lambda$ and $\xi\in\mathbb{C}^{\ast}$ with $\xi^{p_\ell}=1$ such that
$$
\alpha(\tau)=\alpha_\ell(\xi\tau^{p/p_\ell});
$$
the set of such $\ell\in\Lambda$ is denoted by $\Lambda_\alpha$.
\item
$R_\alpha$ is a regular holonomic $\mathbb C[[\tau]]\langle\partial_\tau\rangle$-module, $R_\alpha=R_\alpha[\tau^{-1}]\neq0$ and the rank of $R_\alpha$ is $\sum_{\ell\in\Lambda_\alpha}m_\ell$.
\end{itemize}
\end{theorem}

But we can also compute the characteristic polynomial of the monodromy of $R_{\alpha}$ under the assumption $(\ast)$: for any $\ell,\ell^{'}\in\Lambda$ and for any $\xi_\ell,\xi_{\ell^{'}}\in\mathbb{C}^{\ast}$ with $\xi_\ell^{p_\ell}=1$ and $\xi_{\ell^{'}}^{p_{\ell^{'}}}=1$,
$$(\ell,\xi_\ell)\neq(\ell^{'},\xi_{\ell^{'}})\Longrightarrow \alpha_\ell(\xi_\ell\tau^{\sfrac{p}{p_\ell}})+\delta_{\ell}(0)\neq\alpha_{\ell^{'}}(\xi_{\ell^{'}}\tau^{\sfrac{p}{p_{\ell^{'}}}})+\delta_{\ell^{'}}(0).$$

We remark that $\rho^{'}=(\rho,id):D^{'}\times\mathbb{P}^{1}\to D\times\mathbb{P}^1$ is a normalization of all the $S_{\ell}$'s. For any $\ell\in\Lambda$, $\rho^{'-1}(S_{\ell})$ is the union of some smooth irreductible analytic curves $S_{\ell}^i$.

Then to any $S_{\ell}$ we associate $\zeta_{\ell}\in\mathbb{C}[\lambda]$: the characteristic polynomial of the monodromy of $\Phi_{h_{\ell}}(DR~\mathcal{M})$ around $(0,0)$ in a normalization of $S_{\ell}$, where $h_{\ell}=0$ is an equation of $S_{\ell}$. Here, $\Phi_{h_{\ell}}$ is the vanishing cycle functor along $S_{\ell}$ for bounded complex with constructible cohomology (cf. \cite{MeMai}, $\S 1$) and $\DR$ is the de Rham functor (cf. Definition $2.6.4$ p. $27$ in \cite{Me}). $\Phi_{h_{\ell}}(DR~\mathcal{M})$ has support included in $S_{\ell}$ and $\Phi_{h_{\ell}}(DR~\mathcal{M})_{|S_{\ell}\setminus\{(0,0)\}}$ is just a local system. Then, $\zeta_{\ell}$ is the characteristic polynomial of the monodromy of the local system  $\rho^{-1}(\Phi_{h_{\ell}}(DR~\mathcal{M}))_{|S_{\ell}^i\setminus\{(0,0)\}}$ around $(0,0)$ (do not confuse with the monodromy around $S_{\ell}$). As $S_{\ell}^i\simeq S_{\ell}^j$, for $i,j\in\{1,\ldots,p_{\ell}\}$, $\zeta_{\ell}$ is independant of $i$.

\begin{theorem}\label{zeta}
Under the assumption $(\ast)$, the characteristic polynomial of the monodromy of $R_\alpha$ is equal to $\prod_{\ell\in\Lambda_\alpha}\zeta_{\ell}$.
\end{theorem}

Conversely, we prove in Theorem \ref{realisation} that given any formal holonomic $\mathbb{C}[[t]]\langle\partial_t\rangle$-module $\widehat{\mathcal{N}}$, there exist a ramification $\rho:\tau\to t=\tau^p$ and a regular holonomic $\mathcal D_{D\times\mathbb P^1}$-module $\mathcal{M}$, such that $\rho^{\ast}(\widehat{\mathcal{N}})$ is isomorphic to the formalization of $\rho^{\ast}(\mathcal H^0p_{1+}(\mathcal{M}e^{p_2}))_0$.

In the algebraic setting, let us assume that $U$ is affine and that $f$ and $g$ are regular functions on $U$. Let $\mathcal{M}$ be a holonomic $\mathcal{D}^{\mathrm{alg}}_U$-module which has regular singularities included at infinity. The direct images $\mathcal{N}^k:=\mathcal{H}^k f_+(\mathcal{M}e^g)$ are holonomic $\mathbb C[t]\langle\partial_t\rangle$-modules with singularities at finite distance on the affine line $\mathbb{A}^1$ and at infinity. We will reduce the computation of the formal invariants of $\mathcal{N}^k$ at each of its singularities $c\in\mathbb{A}^1\cup\{\infty\}$ to the situation of Theorem \ref{1} through the diagram
\[
U\xrightarrow{~(f,g)~}\mathbb{A}^1\times\mathbb{A}^1\stackrel{i}{\to} \mathbb{P}^1\times\mathbb{P}^1\stackrel{i_c}{\leftarrow} D_c\times\mathbb{P}^1.
\]
Let $\mathcal{P}_c$ denote the direct image $i_c^{\ast}(i_+(f,g)_+\mathcal{M})^{\mathrm{an}}$. It consists in a complex of $\mathcal{D}_{D_c\times\mathbb{P}^1}$-modules with regular holonomic cohomology modules $\mathcal{H}^k\mathcal{P}_c$.
\begin{theorem}\label{2}
The germs $\mathcal{N}^k_c$ and $\mathcal{H}^0p_{1+}(\mathcal{H}^k\mathcal{P}_ce^{p_2})_c$ have the same formal irregular part.
\end{theorem}

\begin{figure}[h]
\includegraphics[height=2.5cm]{Zlalg.eps}\\
Singular support of $\mathcal{H}^k(i_+(f,g)_+\mathcal{M})^{\mathrm{an}}$.
$c'$ is a regular singularity of $\mathcal{N}^k$. $c$ and $\infty$ are irregular singularities.
\end{figure}

\begin{example}
Let $U$ be a smooth affine surface. Suppose $f,g:U\to\mathbb{A}^1$ are algebraically independant and $\mathcal{M}=\mathcal{O}_U^{\mathrm{alg}}$. The only interesting cohomology module to consider is $\mathcal{N}=\mathcal{H}^0f_+(\mathcal{O}_U^{\mathrm{alg}}e^g)$; the others cohomology modules have punctual support (cf. Proposition 5.1 of \cite{Ro1}).

Let $c\in\mathbb{A}^1\cup\{\infty\}$ and $S$ be the singular support of $\mathcal{H}^0i_+(f,g)_+\mathcal{O}_U$ in the neighbourhood of $(c,\infty)$. Theorem \ref{1} and Theorem \ref{2} allow us to describe the Newton polygon of $\mathcal{N}_c$, up to a suitable translation, using the set $\{S_{\ell}~|~\ell\in\Lambda\}$ of irreducible components of $S$  which are distinct of $\{c\}\times\mathbb{P}^1$ or $\mathbb{P}^1\times\{\infty\}$ and the multiplicity $m_{\ell}$ of $T^{\ast}_{S_{\ell}}(\mathbb{P}^1\times\mathbb{P}^1)$ in the characteristic cycle of $\mathcal{H}^0i_+(f,g)_+\mathcal{O}_U$.

But we can give a geometrical interpretation of the $S_{\ell}$'s and the $m_{\ell}$'s.
Let $\mathbb{X}$ be a smooth compactification of $U$ such that there exists two holomorphic functions $F,G:\mathbb{X}\to\mathbb{P}^1$  which extend $f$ and $g$. Let $\Gamma$ be the critical locus of $(F,G)$ and $D$ be the divisor $\mathbb{X}\setminus U$.
We denote by $\Delta_1$ (resp. $\Delta_2$) the cycle in $\mathbb{P}^1\times\mathbb{P}^1$ which is the closure of $(F,G)(\Gamma)\cap (\mathbb{A}^1\times\mathbb{A}^1)$ (resp. $(F,G)(D)\cap (\mathbb{A}^1\times\mathbb{A}^1)$), where the image is counted with multiplicity. If $c\in\mathbb{A}^1$ (resp. $c=\infty$), the $S_{\ell}$'s are the irreducible components of the germ $\Delta_1\setminus(\{c\}\times\mathbb{P}^1)$ (resp. $\Delta_1\cup\Delta_2$) at $(c,\infty)$; $m_{\ell}$ is the multiplicity of $S_{\ell}$ in $\Delta_1\setminus(\{c\}\times\mathbb{P}^1)$ (resp. $\Delta_1\cup\Delta_2$). This computation can be found in \cite{Ro2} (Theorem 6.4.1).
\end{example}

We shall assume that the reader is familiar with the definitions and the properties of the specialization functor and the nearby cycles functor for $\mathcal D$-modules along a hypersurface and we refer to \cite{Ka}, \cite{Ma2}, \cite{MeSa}, \cite{LaMa} and \cite{MeMai} for details. Given $Y\subset X$ a smooth hypersurface of $X$ defined by a global equation $h=0$ and given $\mathcal{M}$ a specializable $\mathcal{D}_X$-module, we will denote by $\spe_Y\mathcal{M}$ the specialization of $\mathcal{M}$ along $Y$ and by $\Psi_h\mathcal{M}$ the nearby cycle module of $\mathcal{M}$ along $Y$. If $V_{\bullet}\mathcal{M}$ is the canonical V-filtration with respect to the lexical order on $\mathbb{C}\simeq\mathbb{R}\oplus i\mathbb{R}$, $\spe_Y\mathcal{M}=\oplus_{\beta\in\mathbb{C}}\sfrac{V_{\beta}\mathcal{M}}{V_{<\beta}\mathcal{M}}$ and
$\Psi_h\mathcal{M}=\oplus_{-1\leq\beta<0}\sfrac{V_{\beta}\mathcal{M}}{V_{<\beta}\mathcal{M}}$. Same definitions hold for $Y$  hypersurface with normal crossings.
We will also denote by $\Psi_h$ the nearby cycle functor for bounded complexes with constructible cohomology (cf \cite{MeMai} \S 1).

Let us explain the method we use to compute the formal invariants. We note that if a summand $R_\alpha e^\alpha$ shows up in the formal decomposition of a $\mathbb{C}\{t\}\langle\partial_t\rangle$-module $\mathcal{N}_0$ after a ramification $\rho$, then $R_\alpha$ is the formal regular part of $((\rho^*\mathcal{N})e^{-\alpha})_0$. This formal regular part can be recovered from the specialization of $(\rho^*\mathcal{N})e^{-\alpha}$ at the origin (cf. Example 5.2.1 in \cite{LaMa}). Moreover, as $R_{\alpha}$ is localized at the origin, the rank of $R_{\alpha}$ and the characteristic polynomial of the monodromy of $R_{\alpha}$ can be computed using the nearby cycles module of $(\rho^*\mathcal{N})e^{-\alpha}$ at the origin.
One of the main results we use is the commutation of the specialization functor (resp. nearby cycles functor) with proper direct images (cf. Theorem $9.4.1$ in \cite{LaMa} for complexes of $\mathcal{D}$-modules and Theorem $4.8-1$ in \cite{MeSa} for $\mathcal{D}$-modules). This property reduces the proof of Theorem \ref{1} to local analytic computations on a suitable blowing-up space of $D\times\mathbb P^1$ above $(0,\infty)$. It is also the main argument in the proof of Theorem \ref{2}.

\section{Direct image of holonomic $\mathcal{D}$-module of exponential type: the case of projections}

This section is devoted to the proof of Theorems \ref{1} and \ref{zeta}. We first note that Theorem \ref{1} $(2)$ induces the construction of the Newton polygon (Theorem \ref{1} $(1)$).

Let $s\in\mathbb{N}^{\ast}$. Denote by $\Gamma_s$ the set of $\alpha\in\Gamma$ with pole of order $s$ and by $\Lambda_s$ the set of $\ell\in\Lambda$ such that $\sfrac{pq_{\ell}}{p_{\ell}}=s$. The Newton polygon of $\oplus_{\alpha\in\Gamma_s}R_{\alpha}e^{\alpha}$ has just one slope, equal to $s$; its height is the product of the rank of $\oplus_{\alpha\in\Gamma_s}R_{\alpha}$ by the slope. According to Theorem \ref{1} $(2)$, the slope is equal to $\sfrac{pq_{\ell}}{p_{\ell}}$, for all $\ell\in\Lambda_s$, and the height is  $\sum_{\alpha\in\Gamma_s}\sum_{\ell\in\Lambda_{\alpha}}\sfrac{pq_{\ell}m_{\ell}}{p_{\ell}}=\sum_{\ell\in\Lambda_s}pq_{\ell}m_{\ell}$. If $\widetilde{Q_{\ell}}$ denotes the convex hull in $\mathbb{R}^2$ of the union of $Q$ and $(m_{\ell}p_{\ell},pm_{\ell}q_{\ell})+Q$, the Newton polygon of $\rho^{\ast}\mathcal{N}_0$ is, after a suitable translation, the Minkowski sum $\sum_{\ell\in\Lambda}\widetilde{Q_{\ell}}$. We deduce the Newton polygon of $\mathcal{N}_0$ from the one of $\rho^{\ast}\mathcal{N}_0$ by a dilation of the vertical axis in a ratio $\sfrac{1}{p}$ (cf. Lemma 5.4.3 p. $34$ of \cite{Sa}).

We focus now on the proof of Theorem \ref{1} $(2)$ and Theorem \ref{zeta}.

\subsection{Ramification}

The choice of the ramification enables us to reduce the proof of Theorem \ref{1} and \ref{zeta} to the case where the $p_l$'s are equal to $1$; it is the non ramified case ($p=1$).

We begin with a base change formula. Let $p_1^{'}:D^{'}\times\mathbb{P}^1\to D^{'}$ and $p_2^{'}:D^{'}\times\mathbb{P}^1\to\mathbb{P}^1$ be the canonical projections. Consider the cartesian diagram:
$$\xymatrix{
D^{'}\times\mathbb{P}^1\ar[r]^{\rho^{'}}\ar[d]^{p_1^{'}}&D\times\mathbb{P}^1\ar[d]^{p_1}\\
D^{'}\ar[r]^{\rho}&D.}$$

\begin{lemma}\label{comm}
$\rho^{\ast}p_{1+}(\mathcal{M}e^{p_2})=p_{1+}^{'}(\rho^{'\ast}(\mathcal{M})e^{p_2^{'}})$.
\end{lemma}

\begin{proof}
Denote by $~^p\DR_{D\times\mathbb{P}^1/D}$ (resp. $~^p\DR_{D^{'}\times\mathbb{P}^1/D^{'}}$) the relative de Rham functor of $p_1$ (resp. $p_1^{'}$). We adopt the convention that the relative de Rham complexes are concentrated in negative degrees. We have isomorphisms of complexes of $\mathcal{D}_{D^{'}}$-modules:
$$
\begin{array}{ll}
\rho^{\ast}p_{1+}&(\mathcal{M}e^{p_2})=\\
&=\mathcal{D}_{D^{'}\to D}\otimes_{\rho^{-1}\mathcal{D}_D}\rho^{-1}\mathbb{R}p_{1\ast}
~^p\DR_{D\times\mathbb{P}^1/D}(\mathcal{M}e^{p_2})\text{ (by definition),}\\
&=\mathcal{D}_{D^{'}\to D}\otimes_{\rho^{-1}\mathcal{D}_D}\mathbb{R}p^{'}_{1\ast}\rho^{'-1}
~^p\DR_{D\times\mathbb{P}^1/D}(\mathcal{M}e^{p_2}),\\
&\text{(Proposition 2.6.7 in \cite{KaSc}),}\\
&=\mathbb{R}p^{'}_{1\ast}(\mathcal{D}_{D^{'}\times\mathbb{P}^1\to D\times\mathbb{P}^1}\otimes_{\rho^{'-1}\mathcal{D}_{D\times\mathbb{P}^1}}\rho^{'-1}
~^p\DR_{D\times\mathbb{P}^1/D}(\mathcal{M}e^{p_2})),\\
&\text{ (Proposition 2.6.6 in \cite{KaSc}),}\\
&=\mathbb{R}p^{'}_{1\ast}~^p\DR_{D^{'}\times\mathbb{P}^1/D^{'}}((\mathcal{D}_{D^{'}\times\mathbb{P}^1\to D\times\mathbb{P}^1}\otimes_{\rho^{'-1}\mathcal{D}_{D\times\mathbb{P}^1}}\rho^{'-1}
(\mathcal{M}))e^{p_2^{'}}),\\
&=p_{1+}^{'}(\rho^{'\ast}(\mathcal{M})e^{p_2^{'}}).
\end{array}$$

\end{proof}

We deduce that $\rho^{\ast}\mathcal{N}=\mathcal{H}^0p_{1+}^{'}(\rho^{'\ast}(\mathcal{M})e^{p_2^{'}})$.

Then we remark that the set $\{S_{\ell}^i~|~\ell\in\Lambda,~i=1,\ldots,p_{\ell}\}$ consists of the irreducible components of the singular support $S^{'}$ of $\rho^{'\ast}(\mathcal{M})$ in a neighbourhood $U^{'}$ of $(0,\infty)$, which are distinct of $\{0\}\times\mathbb{P}^1$ or $D^{'}\times\{\infty\}$. Let $\xi_1,\ldots,\xi_{p_{\ell}}$ be the $p_{\ell}$-roots of the unity. Up to change of indices $i$, a parametrization of $S_{\ell}^i$ is given by $\gamma(\tau)=(\tau,\alpha_{\ell}(\xi_i\tau^{\sfrac{p}{p_{\ell}}})+\delta_{\ell}(\xi_i\tau^{\sfrac{p}{p_{\ell}}}))$; in particular, the intersection multiplicity
$p_{\ell}^i$ at $(0,\infty)$ of $S_{\ell}^i$ with $\{0\}\times\mathbb{P}^1$ is $1$.

In the next section, we will prove Theorem \ref{1} $(2)$ and Theorem \ref{zeta} in the case where all the $p_{\ell}$'s are equal to $1$. Then, we may apply it to $\rho^{'\ast}(\mathcal{M})$. At last, we have to compute the data associated with each $S_{\ell}^i$ using data associated with the $S_{\ell}$'s.
But we have:\\

$\bullet$ the polar part of a Puiseux parametrization of $S_{\ell}^i$ at $(0,\infty)$ is $\alpha_{\ell}(\xi_i\tau^{\sfrac{p}{p_{\ell}}})$. Moreover, if the assumption $(\ast)$ is fulfilled for $\mathcal{M}$ it is also fulfilled for $\rho^{'\ast}(\mathcal{M})$, \\

$\bullet$ $m_{\ell}$ is the multiplicity of  $T^{\ast}_{S_{\ell}^i}(D^{'}\times\mathbb{P}^1)$ in the characterstic cycle of $\rho^{'\ast}(\mathcal{M})$,\\

$\bullet$ $\zeta_{\ell}$ is the characteristic polynomial of the monodromy of the local system  $\Phi_{h_{\ell}^i}\DR~(\rho^{'\ast}(\mathcal{M}))_{|S_{\ell}^i\setminus\{(0,0)\}}$ around $(0,0)$, where $h_{\ell}^i=0$ is an equation of $S_{\ell}^i$.

\subsection{Non ramified case: $p_{\ell}=1$, for all $\ell\in\Lambda$}
If we prove that the rank of $R_{\alpha}$ is equal to $\sum_{\ell\in\Lambda_{\alpha}}m_{\ell}$, the irregularity number at $0$ of $\mathcal{N}$ and of $\oplus_{\alpha\in\Gamma}R_{\alpha}e^{\alpha}$ are both equal to $\sum_{\ell\in\Lambda}m_{\ell}q_{\ell}$ (cf. \cite{Ro} for $\mathcal{N}$). Then it is sufficient to prove that each summand $R_{\alpha}e^{\alpha}$, with $\alpha\in\Gamma$, shows up in the formal decomposition of $\mathcal{N}_0$ and to compute the rank and the characteristic polynomial of the monodromy of each $R_{\alpha}$.

Let $\alpha\in\Gamma$. We denote by $\spe_0(\mathcal{N}e^{-\alpha})$ the specialization of $\mathcal{N}e^{-\alpha}$ at the origin. $\spe_0(\mathcal{N}e^{-\alpha})_0$ is a $\mathbb{C}[t]\langle\partial_t\rangle$-module. As announced in the introduction, $R_{\alpha}$ is the formal regular part of $(\mathcal{N}e^{-\alpha})_0$. It can be recovered using the isomorphism $R_{\alpha}=\mathbb{C}[[t]]\otimes_{\mathbb{C}[t]}\spe_0(\mathcal{N}e^{-\alpha})_0$ (cf. Example $5.2.1$ of \cite{LaMa}).
Let $\Psi_t(\mathcal{N}e^{-\alpha})$ be the nearby cycles module of $\mathcal{N}e^{-\alpha}$ at the origin. $\Psi_t(\mathcal{N}e^{-\alpha})_0$ is a finite dimensional $\mathbb{C}$-vector space equipped with an endomorphism of monodromy induced by $\exp(-2i\pi t\partial_t)$.
As $\mathcal{N}e^{-\alpha}$ is localized at the origin, the rank of $R_{\alpha}$ is the dimension of $\Psi_t(\mathcal{N}e^{-\alpha})_0$ and the characteristic polynomials of their monodromies are equal.

But we have, $\Psi_t(\mathcal{N}e^{-\alpha})_0=R\Gamma(\{0\}\times\mathbb{P}^1,\DR~\Psi_{p_1}(\mathcal{M}e^{p_2-\alpha\circ p_1})[+1])$. Indeed,

$$\begin{array}{ll}
\mathcal{N}e^{-\alpha}&=\mathcal{O}_D[\frac{1}{t}]e^{-\alpha}\otimes_{\mathcal{O}_D}\mathcal{H}^0p_{1+}(\mathcal{M}e^{p_2}),\\
&=\mathcal{O}_D[\frac{1}{t}]e^{-\alpha}\otimes^{\mathbb{L}}_{\mathcal{O}_D}\mathcal{H}^0p_{1+}(\mathcal{M}e^{p_2}),\text{ (flatness),}\\
&=(\mathcal{O}_D[\frac{1}{t}]e^{-\alpha}\otimes_{\mathcal{O}_D}\mathcal{D}_D)\otimes^{\mathbb{L}}_{\mathcal{D}_D}\mathcal{H}^0p_{1+}(\mathcal{M}e^{p_2}),\\
&=\mathcal{H}^0p_{1+}((\mathcal{O}_{D\times\mathbb{P}^1}[\frac{1}{t}]e^{-\alpha\circ p_1}\otimes_{\mathcal{O}_{D\times\mathbb{P}^1}}\mathcal{D}_{D\times\mathbb{P}^1})\otimes_{\mathcal{D}_D}^{\mathbb{L}}\mathcal{M}e^{p_2}),\\
&\text{ (Prop 2.6.6 in \cite{KaSc}),}\\
&=\mathcal{H}^0p_{1+}(\mathcal{M}e^{p_2-\alpha\circ p_1}).
\end{array}$$

Let $\overline{p_1}:\{0\}\times\mathbb{P}^1\to\{0\}$. According to the commutation of the nearby cycle functor with proper direct image (cf. Theorem 4.8-1 p. $226$ of \cite{MeSa}),
$$\begin{array}{ll}
\Psi_t(\mathcal{N}e^{-\alpha})_0&=\mathcal{H}^0\overline{p_1}_+(\Psi_{p_1}(\mathcal{M}e^{p_2-\alpha\circ p_1}))_0,\\
&=R\Gamma(\{0\}\times\mathbb{P}^1,\DR~\Psi_{p_1}(\mathcal{M}e^{p_2-\alpha\circ p_1})[+1]), \text{ (cf. \cite{Ma3} p.$5$).}
\end{array}$$

Then, Theorem \ref{1} $(2)$ and Theorem \ref{zeta} follows from the proposition:
\begin{proposition}\label{11}
$\Psi_{p_1}(\mathcal{M}e^{p_2-\alpha\circ p_1})$ has support included in $(0,\infty)$ and the Euler characteristic of  $\DR~\Psi_{p_1}(\mathcal{M}e^{p_2-\alpha\circ p_1})_{(0,\infty)}$ is $-\sum_{\ell\in\Lambda_{\alpha}}m_{\ell}$.\\
Moreover, under the assumption $(\ast)$, the zeta function of its monodromy is $\prod_{\ell\in\Lambda_\alpha}\zeta_{\ell}^{-1}$.
\end{proposition}

The proof is given in the next sections.

\begin{remark}
In the non ramified case, the assumption $(\ast)$ is equivalent to:
$$\forall\ell,\ell^{'}\in\Lambda_{\alpha},~\ell\neq\ell^{'}\Longrightarrow\delta_\ell(0)\neq\delta_{\ell^{'}}(0).$$
\end{remark}

\subsection{Local computations of nearby cycles modules}
In this section, we prepare the proof of Proposition \ref{11} by giving some computations of nearby cycles modules along a normal crossing. Let $V$ be a neighbourhood of $(0,0)$ in $\mathbb{C}^2$ and $(u,v)$ be some coordinates on $V$. Let $\mathcal{M}$ be a regular holonomic $\mathcal{D}_V$-module and $m,n,k,l\in\mathbb{N}$. We are interested in nearby cycles modules of the type $\Psi_{u^mv^n}(\mathcal{M}e^{\sfrac{1}{u^kv^l}})$.

\subsubsection{Nearby cycles module along $u^mv^n=0$ of a module of exponential type}\label{nota}
Suppose that the singular support of $\mathcal{M}$ is included in $uv=0$. Let $V^{\ast}=V\setminus\{uv=0\}$. $\mathcal{M}_{|V^{\ast}}$ is a holomorphic connection; we denote by $r$ its rank. Thus $(\DR~\mathcal{M})_{|V^{\ast}}$ is a local system. For $P\in V^{\ast}$, we consider the monodromy $T_P$ of $(\DR~\mathcal{M})_P$ around $u=0$. As $V^{\ast}$ is connected and the singular support of $\mathcal{M}$ is included in a normal crossing, the characteristic polynomial of $T_P$ does not depend on $P$ (cf. I.2.2 p. 55 in \cite{MeNa}). We denote it by $\zeta_r$.

\begin{lemma}\label{B}
\begin{enumerate}
\item If $k,m\geq1$, $\Psi_{u^m}(\mathcal{M}e^{\sfrac{1}{u^k}})=0$.
\item If $k,l,m,\geq1$ and $n\geq 0$, $\Psi_{u^mv^n}(\mathcal{M}e^{\sfrac{1}{u^kv^l}})=0$.
\item If $n\geq 1$, the Euler characteristic of  $\DR~\Psi_{uv^n}(\mathcal{M}[\frac{1}{uv}]e^{\sfrac{1}{v}})_{(0,0)}$ is equal to $-r$ and the zeta function of its monodromy is $\zeta_r^{-1}$.
\end{enumerate}
\end{lemma}

\begin{proof}
The proof of the last two points is given in Lemma $4.5.10$ of \cite{Sa2} and its proof. The shift in the third point comes from the convention on the de Rham complexes. The first point is also proved in \cite{Sa2} in the case where $\mathcal{M}=\mathcal{M}[\frac{1}{v}]$. To conclude, it is sufficient to prove it in the case where $\mathcal{M}$ has support in $v=0$. We have $\mathcal{M}=i_+Li^{\ast}(\mathcal{M})[+1]$, where $i:\{v=0\}\hookrightarrow V$,
and $Li^{\ast}(\mathcal{M})[+1]$ is just a module.
As $\Psi_{u^m}(Li^{\ast}(\mathcal{M})[+1]e^{\sfrac{1}{u^k}}))=0$ (computation in one variable), we deduce that $\Psi_{u^m}(\mathcal{M}e^{\sfrac{1}{u^k}})=0$.
\end{proof}

\subsubsection{Nearby cycles module along $u=0$ of a regular $\mathcal{D}_V$-module}

Let $\widetilde{S}$ be the singular support of $\mathcal{M}$ and $\{\widetilde{S_{\ell}}~|~\ell\in\Lambda\}$ be the set consisting of the irreducible components of $\widetilde{S}$ which are distinct of $u=0$. We will assume that {\bf the intersection
multiplicity at $(0,0)$ of $\widetilde{S_{\ell}}$ with $u=0$ is equal to $1$}.

We denote by $r$ the rank of $\mathcal{M}$. To any $\widetilde{S_{\ell}}$, we associate $m_{\ell}$, the multiplicity of the conormal space $T^{\ast}_{\widetilde{S_{\ell}}}V$ in the characteristic cycle of $\mathcal{M}$.

\begin{lemma}\label{A}
The Euler characteristic of $\DR~\Psi_u(\mathcal{M}[\frac{1}{u}])_{(0,0)}$ is equal to  $r-\sum_{\ell\in\Lambda}m_{\ell}$.
\end{lemma}

\begin{proof}
As $\mathcal{M}$ is regular holonomic,
$$\begin{array}{ll}\DR~\Psi_u(\mathcal{M}[\frac{1}{u}])&=\DR~\Psi_u(\mathcal{M})\text{ (Proposition 2.4-3 in \cite{MeMai}),}\\
&=\Psi_u(\DR~\mathcal{M})\text{ (Theorem $4.10.1$ p. 233 in \cite{MeSa}).}\end{array}$$

Let $\epsilon$ and $\eta$ small enough such that any $\widetilde{S_{\ell}}$ intersects
$X_{\epsilon,\eta}=B(0,\epsilon)\cap\{u=\eta\}$ at a unique point $P_{\ell}$ (by assumption) and
$\Psi_u(\DR~\mathcal{M})_{(0,0)}=\mathbb{R}\Gamma(X_{\epsilon,\eta},\DR~\mathcal{M})$ (cf. Remark $1.1-7$ in \cite{MeMai}).
Denote by
$X_{\epsilon,\eta}^c=X_{\epsilon,\eta}\setminus\{P_{\ell},~\ell\in\Lambda\}$.  Let $B_{\ell}$, $\ell\in\Lambda$, be some disjoint balls centered at $P_{\ell}$ of radius small enough.

 According to the index theorem of Kashiwara (Theorem 6.3.1 in \cite{Ka1}), the Euler characteristic of $(\DR~\mathcal{M})_P$, $P\in X_{\epsilon,\eta}^c$, (resp. $(\DR~\mathcal{M})_{P_{\ell}}$) is equal to $r$ (resp. $r-m_{\ell}$).
As $(\DR~\mathcal{M})_{|X_{\epsilon,\eta}}$ is constructible
with respect to the stratification $\{X_{\epsilon,\eta}^c,
P_{\ell},~\ell\in\Lambda\}$, the Euler characteristic of  $\mathbb{R}\Gamma(X_{\epsilon,\eta},\DR~\mathcal{M})$ may be computed using the Mayer-Vietoris Theorem. We obtain:
$$\begin{array}{lll}
\chi(\mathbb{R}\Gamma(X_{\epsilon,\eta},\DR~\mathcal{M}))&=&
\chi(\mathbb{R}\Gamma(X_{\epsilon,\eta}^c,\DR~\mathcal{M}))+\sum_{\ell\in\Lambda}\chi((\DR~\mathcal{M})_{P_{\ell}})\\
&&-\sum_{\ell\in\Lambda}\chi(\mathbb{R}\Gamma(B_{\ell}\setminus\{P_{\ell}\},\DR~\mathcal{M})),\\
&=&\chi(X_{\epsilon,\eta}^c).r+\sum_{\ell\in\Lambda}(r-m_{\ell})\\
&&-\sum_{\ell\in\Lambda}\chi(B_{\ell}\setminus\{P_{\ell}\}).r,\\
&=&r-\sum_{\ell\in\Lambda}m_{\ell}.
\end{array}$$
\end{proof}

The zeta function of the monodromy of $\DR~\Psi_u(\mathcal{M}[\frac{1}{u}])_{(0,0)}$ can not be compute with this method in the general case. Nevertheless, we can do it when the singular support of $\mathcal{M}$ is included in the normal crossing $uv=0$. We adopt the notations of the section \ref{nota}. Let $V_0=\{v=0\}\setminus\{(0,0)\}$. We denote by $\zeta$ the characteristic polynomial of the monodromy of the local system $\phi_v(\DR~\mathcal{M})_{|V_0}$ around $(0,0)$.

\begin{lemma}\label{C}
If the singular support of $\mathcal{M}$ is included in a normal crossing $uv=0$, the zeta function of the monodromy of $\DR~\Psi_u(\mathcal{M}[\frac{1}{u}])_{(0,0)}$ is equal to $\zeta_r\zeta^{-1}$.
\end{lemma}

\begin{proof}
We have $\DR~\Psi_u(\mathcal{M}[\frac{1}{u}])_{(0,0)}=\underset{\underset{\epsilon,\eta}{\leftarrow}}{\lim}R\Gamma(X_{\epsilon,\eta},\DR~\mathcal{M})$.

As the singular support of $\mathcal{M}$ is included in $uv=0$ and as $\DR~\mathcal{M}$ is constructible with respect to a stratification such that $uv=0$ is a union of strata, we have $R\Gamma(X_{\epsilon,\eta},\DR~\mathcal{M})=(\DR~\mathcal{M})_P$, where $P\in V_0$ is the intersection point of $X_{\epsilon,\eta}$ with $v=0$. Then, the zeta function of the monodromy of $\DR~\Psi_u(\mathcal{M}[\frac{1}{u}])_{(0,0)}$ coincide with the zeta function of the monodromy of the complex of local systems $(\DR~\mathcal{M})_{|V_0}$ around $(0,0)$.

Consider the triangle of complexes of local systems:
$$(\DR~\mathcal{M})_{|V_0}\to\Psi_v(\DR~\mathcal{M})_{|V_0}\to\phi_v(\DR~\mathcal{M})_{|V_0}\overset{+1}{\to}.$$

As the zeta function of the monodromy of $\Psi_v(\DR~\mathcal{M})_{|V_0}$ around $(0,0)$ is $\zeta_r$, Lemma \ref{C} follows.
\end{proof}

\subsection{Proof of Proposition \ref{11}}~\\
First we have to prove that $\Psi_{p_1}(\mathcal{M}e^{p_2-\alpha\circ p_1})_{(0,c)}=0$, for any $c\neq \infty$. But in a neighbourhood $U$ of $(0,c)$ in $D\times\mathbb{P}^1$, there exist coordinates $(x,y)$ such that $p_1(x,y)=x$, $p_2(x,y)=y$ and $\alpha\circ p_1\sim\frac{1}{x^q}$, for some $q\in\mathbb{N}^{\ast}$. Therefore,  $\Psi_{p_1}(\mathcal{M}e^{p_2-\alpha\circ p_1})_{(0,c)}=\Psi_x(\mathcal{M}_{|U}e^{\frac{1}{x^q}})_{(0,0)}$. As the singular support of $\mathcal{M}_{|U}$ is not necessarily a normal crossing at $(0,0)$, we can not apply directly Lemma \ref{B} 1.

The idea is to use a resolution  $\pi=(\pi_1,\pi_2):\mathbb{X}\to U$, with exceptional locus $E$, such that the singular support of $\pi^{\ast}(\mathcal{M}_{|U})[\ast\pi_1^{-1}(0)]$ has normal crossings in a neighbourhood of any point of $E$. As $\mathcal{M}$ is holonomic and $\pi$ is a proper isomorphism out of $E$,
$\mathcal{M}_{|U}e^{-\frac{1}{x^q}}=\pi_+(\pi^{\ast}(\mathcal{M}_{|U})e^{-\frac{1}{\pi_1^q}})$ (cf. Proposition 7.4.5 of \cite{Me}).
Using the commutation of the proper direct image with $\Psi$ and $\DR$ (Theorem 4.8.1 p. 226 in \cite{MeSa} and Theorem 5.4.3 p. 77 in \cite{Me}), we deduce that
\label{9}
$$(\ast\ast)~\DR ~\Psi_{x}(\mathcal{M}_{|U}e^{-\frac{1}{x^q}})_{(0,0)}=R\Gamma(E,\DR~\Psi_{\pi_1}(\pi^{\ast}(\mathcal{M}_{|U})e^{-\frac{1}{\pi_1^q}})).$$
\begin{itemize}
\item If $P$ is a smooth point of $\pi^{-1}_1(0)$, we have $\Psi_{\pi_1}(\pi^{\ast}(\mathcal{M}_{|U})e^{-\frac{1}{\pi_1^q}})_P=0$ (cf. Lemma \ref{B} 1).
\item If $P$ is a normal crossing point of $\pi^{-1}_1(0)$, according to Lemma \ref{B} 2, $\Psi_{\pi_1}(\pi^{\ast}(\mathcal{M}_{|U})e^{-\frac{1}{\pi_1^q}})_P=0$.
\end{itemize}

Then $\Psi_{x}(\mathcal{M}_{|U}e^{-\frac{1}{x^q}})_{(0,0)}=0$ and $\Psi_{p_1}(\mathcal{M}e^{p_2-\alpha\circ p_1})$ has support included in $(0,\infty)$.

\vspace{0.5cm}

Let us now focus on $\Psi_{p_1}(\mathcal{M}e^{p_2-\alpha\circ p_1})_{(0,\infty)}$. In a neighbourhood $U$ of $(0,\infty)$ in $D\times\mathbb{P}^1$, there exist coordinates $(x,y)$ such that $p_1(x,y)=x$, $p_2(x,y)=\sfrac{1}{y}$ and $p_2-\alpha\circ p_1(x,y)=\sfrac{1}{y}-\alpha(x)$. To simplify notation, we will set $g=\sfrac{1}{y}-\alpha(x)$. Then we have:
$$
\Psi_{p_1}(\mathcal{M}e^{p_2-\alpha\circ p_1})_{(0,\infty)}=\Psi_{x}(\mathcal{M}_{|U}[\frac{1}{xy}]e^{g})_{(0,0)}.$$

We have to compute the Euler characteristic and the zeta function of the monodromy of $\DR~\Psi_{x}(\mathcal{M}_{|U}[\frac{1}{xy}]e^g)_{(0,0)}$.
We can not use directly the local computations because $g$ is not equivalent to a function of the type $\sfrac{1}{x^ky^l}$ at $(0,0)$. But we will be led to this situation with the help of a resolution. The choice of a good resolution is relevant for the end of the proof.

\subsubsection{Choice of a resolution}

\begin{lemma}\label{resolution}
There exists a resolution $\pi=(\pi_1,\pi_2):\mathbb{X}\to U$ with exceptional locus $E$ (actually it is a finite composition of the blow-up of the point $(0,0)\in U$ and some blow-up of points in its exceptional locus) which satisfies the following conditions:
\begin{enumerate}
\item Let $\widetilde{E}=\pi^{-1}(\{xy=0\})$. There exists a unique irreducible component $E_d$ of $E$ which intersects $\overline{\widetilde{E}\setminus E_d}$ in a unique point $P$. Moreover, in a neighbourhood of any point $Q\in E$, we can choose local coordinates $(u,v)$ on $\mathbb{X}$ such that:
\begin{enumerate}
\item If $Q\notin E_d$,
\begin{itemize}
\item $\pi_1(u,v)=u^m$ and $g\circ\pi(u,v)\sim\sfrac{1}{u^k}$, with $m,k\geq 1$,

or
\item $\pi_1(u,v)=u^mv^n$ and $g\circ\pi(u,v)\sim\sfrac{1}{u^kv^l}$, with $k,l,m\geq 1$, $n\geq 0$.
\end{itemize}
\item If $Q=P$,
\begin{itemize}
\item $u=0$ (resp. $v=0$) is an equation of $E_d$ (resp. of the other component of $E$),
\item $\pi_1(u,v)=uv^n$ and $g\circ\pi(u,v)\sim\sfrac{1}{v}$, with $n\geq 1$.
\end{itemize}
\item If $Q\in E_d\setminus\{P\}$,
\begin{itemize}
\item $u=0$ is an equation of $E_d$,
\item $\pi_1(u,v)=u$ and $g\circ\pi(u,v)=a+v$, with $a\in\mathbb{C}$.
\end{itemize}
\end{enumerate}
\item The singular support of $\pi^{\ast}(\mathcal{M}_{|U}[\frac{1}{xy}])=\pi^{\ast}(\mathcal{M}_{|U})[\ast\widetilde{E}]$ has
at most normal crossings at any point of
$\overline{E\setminus E_d}$.
\end{enumerate}
\end{lemma}

\newpage

\begin{figure}[h]
\includegraphics[height=3cm]{exclocus.eps}
\caption{Exceptional locus $E$}
\end{figure}

\begin{proof}
By reducing to the same denominator, we have $g=\sfrac{(x^q-y\beta)}{x^qy}$, with $\beta\in\mathbb{C}[x]$ and $\beta(0)\neq 0$.

Let us begin by proving the first point. We will perform $2q$ blow-up of points. The resolution tree we will obtain has just one branch and the component $E_d$ will be the exceptional locus of the last blow-up.

The first $q$ blow-up allow us to decrease the degree on $x$ in the numerator. We obtain a resolution $\pi=(\pi_1,\pi_2)$ composition of $q$ blow-up
$$
\begin{array}{ccccccc}
X^{(q)}&\overset{\pi^{(q)}}{\to}&\ldots&\to&X^{(1)}&\overset{\pi^{(1)}}{\to}&U\\
\cup&&&&\cup&&\\
E_q&&&&E_1&&
\end{array},$$
with exceptional locus $E=E_1\cup\ldots\cup E_q$, such that $g\circ\pi$ can be written locally as in the case (a) except at a point $I\in E_q\setminus E_{q-1}$. In a neighbourhood of this point, we can choose local coordinates $(u,v)$ such that $\pi_1(u,v)=u$ and $g\circ\pi(u,v)=\sfrac{(u\gamma(u)-v\beta(u))}{u^qh(v)}$, with $h(v)=v+\sfrac{1}{\beta(0)}$ and $\gamma(u)=\sfrac{(\beta(0)-\beta(u))}{\beta(0)u}$.

Now the $q-1$ next blow-up allow us to decrease the degree in $u$ in the denominator and the degree in $u$ of $\gamma(u)$. We obtain a new resolution $\pi=(\pi_1,\pi_2)$, composition of $2q-1$ blow-up, such that $g\circ\pi$ can be written locally as in the case (a) except at a point $J\in E_{2q-1}\setminus E_{2q-2}$. In some suitable local coordinates $(u,v)$ in a neighbourhood of $J$,  $\pi_1(u,v)=u$ and $g\circ\pi(u,v)=\sfrac{(cu-v\beta(u))}{uh(u,v)}$, with $c$ constant and $h(u,v)$ invertible.

At last, we blow-up $J$. The exceptional locus of this last blow up is $E_d$ and we can verify the local behaviour of $g$ on $E_d$. The details are left to the reader.

To prove the second point, it is sufficient to perform additional blow-up of points in $\overline{E\setminus E_d}$.
\end{proof}

We denote by $\widetilde{S_{\ell}}$ the strict transform of $S_{\ell}$.
\begin{corollary}\label{stricttransform}\mbox{}\par
$(1)$ Let $\ell\in\Lambda$. $\widetilde{S_{\ell}}\cap E_d\neq\emptyset$ if
and only if $\ell\in\Lambda_{\alpha}$.

$(2)$ The assumption $(\ast)$ is equivalent to:
$\forall \ell,\ell^{'}\in\Lambda_{\alpha},~\ell\neq\ell^{'}\Longrightarrow\widetilde{S_{\ell}}\cap E_d\neq\widetilde{S_{\ell^{'}}}\cap E_d$.
\end{corollary}

\begin{proof}
$(1)$ According to the choice of the resolution,
$\widetilde{S_{\ell}}\cap E_d\neq\emptyset$ if and only if
$\widetilde{S_{\ell}}\cap (E_d\setminus\{P\})\neq\emptyset$. Let $Q_{\ell}\in\widetilde{S_{\ell}}\cap (E_d\setminus\{P\})$ and  $(u,v)$ be some local coordinates in the
neighbourhood of $Q_{\ell}$ such that $f\circ\pi(u,v)=a+v$ and
$\pi_1(u,v)=u$, with $a\in\mathbb{C}$.

As the
intersection multiplicity of $S_{\ell}$ with $\{0\}\times\mathbb{P}^1$ is $1$ and
as $\pi_1(u,v)=u$, the intersection multiplicity of
$\widetilde{S_{\ell}}$ with $E_d$ is also $1$. Then:
\begin{displaymath}
\begin{split}
&\widetilde{S_{\ell}}\cap E_d\neq\emptyset,\\
&\Longleftrightarrow\exists\delta(s)\in\mathbb{C}\{s\}\text{ such that }a+v=\delta(u) \text{ is an equation of }\widetilde{S_{\ell}},\\
&\Longleftrightarrow\exists\delta(s)\in\mathbb{C}\{s\}\text{ such that }f\circ\pi(u,v)=\delta\circ\pi_1(u) \text{ is an equation of }\widetilde{S_{\ell}},\\
&\Longleftrightarrow\exists\delta(s)\in\mathbb{C}\{s\}\text{ such that }f(x,y)=\delta(x) \text{ is an equation of }S_{\ell},\\
&\Longleftrightarrow\exists\delta(s)\in\mathbb{C}\{s\}\text{ such that }\sfrac{1}{y}=\alpha(x)+\delta(x) \text{ is an equation of }S_{\ell},\\
&\Longleftrightarrow \ell\in\Lambda_{\alpha}.
\end{split}
\end{displaymath}

$(2)$ Here, we want to understand the behaviour of $S_{\ell}$ after the resolution $\pi$. We will keep the same notations as in the proof of Lemma \ref{resolution}.

Let $\ell\in\Lambda_{\alpha}$. Then $S_{\ell}$ has equation $h(x,y)=(\beta(x)+x^q\delta_{\ell}(x))y-x^q=0$.

After the first $q$ blow-up, in the neighbourhood of the point $I\in E_q\setminus E_{q-1}$ with local coordinates $(u,v)$,
$$h\circ\pi(u,v)=u^q\big((\beta(u)+u^q\delta_{\ell}(u))v+u(-\gamma(u)+u^{q-1}\sfrac{\delta_{\ell}(u)}{\beta(0)})\big).$$
After the next $(q-1)$ blow-up, in the neighbourhood of the point $J$, we can choose local coordinates $(u,v)$ such that
$$h\circ\pi(u,v)=u^{2q-1}\big((\beta(u)+u^q\delta_{\ell}(u))v+u(-c+\sfrac{\delta_{\ell}(u)}{\beta(0)}+u\widetilde{\delta}(u))\big),$$
where $\widetilde{\delta}(u)\in\mathbb{C}\{u\}$ and $c$ is a constant which depends only on $\alpha$.

At last, we blow-up $J$ and obtain that in the chart $u=s$, $v=st$,
$$h\circ\pi(s,t)=s^{2q}\big((\beta(s)+s^q\delta_{\ell}(s))t-c+\sfrac{\delta_{\ell}(s)}{\beta(0)}+s\widetilde{\delta}(s)\big).$$
Then, we see that the strict transform $\widetilde{S_{\ell}}$ of $S_{\ell}$ interesct $E_d$ at a point which depends only on $\alpha$ and $\delta_{\ell}(0)$. The assertion $(2)$ follows.
\end{proof}

\subsubsection{Use of the local computations}

There exists an isomorphism:
$$\begin{array}{ll}
\DR~\Psi_{x}(\mathcal{M}_{|U}[\frac{1}{xy}]e^g)_{(0,0)}
&=R\Gamma(E,\DR~\Psi_{\pi_1}(\pi^{\ast}(\mathcal{M}_{|U}[\frac{1}{xy}])e^{g\circ\pi})),\\
&\text{(Same proof as for the isomorphism $(\ast\ast)$ p. \pageref{9}),}\\
&=R\Gamma(E,\DR~\Psi_{\pi_1}(\pi^{\ast}(\mathcal{M}_{|U})[\ast\widetilde{E}]e^{g\circ\pi})).
\end{array}$$

Let us first prove that the complex $\DR~\Psi_{\pi_1}(\pi^{\ast}(\mathcal{M}_{|U})[\ast\widetilde{E}]e^{g\circ\pi})_{|E}$ has support included in $E_d$. We have two cases to consider:

\begin{enumerate}
\item In a neighbourhood $V\subset\mathbb{X}$ of a smooth point $Q$ in $E\setminus E_d$, according to Lemma \ref{resolution}, there exist coordinates $(u,v)$ on $V$ such that $\pi_1(u,v)=u^m$, $g\circ\pi(u,v)\sim\sfrac{1}{u^k}$ and the singular support of $\pi^{\ast}(\mathcal{M}_{|U})[\ast\widetilde{E}]$ is included in $uv=0$. Then, according to Lemma \ref{B} 1, $$\Psi_{\pi_1}(\pi^{\ast}(\mathcal{M}_{|U})[\ast\widetilde{E}]e^{g\circ\pi})_Q=\Psi_{u^m}(\pi^{\ast}(\mathcal{M}_{|U})[\ast\widetilde{E}]e^{\frac{1}{u^k}})_{(0,0)}=0.$$

\item In a neighbourhood $V\subset\mathbb{X}$ of a normal crossing point $Q$ in $E\setminus E_d$, according to Lemma \ref{resolution}, there exist coordinates $(u,v)$ on $V$ such that $\pi_1(u,v)=u^mv^n$, $g\circ\pi(u,v)\sim\sfrac{1}{u^kv^l}$ and the singular support of $\pi^{\ast}(\mathcal{M}_{|U})[\ast\widetilde{E}]$ is included in $uv=0$. Then, according to Lemma \ref{B} 2, $$\Psi_{\pi_1}(\pi^{\ast}(\mathcal{M}_{|U})[\ast\widetilde{E}]e^{g\circ\pi})_Q=\Psi_{u^mv^n}(\pi^{\ast}(\mathcal{M}_{|U})[\ast\widetilde{E}]e^{\frac{1}{u^kv^l}})_{(0,0)}=0.$$
\end{enumerate}

Then $\DR~\Psi_{x}(\mathcal{M}_{|U}[\frac{1}{xy}]e^g)_{(0,0)}=R\Gamma(E_d,\DR~\Psi_{\pi_1}(\pi^{\ast}(\mathcal{M}_{|U})[\ast\widetilde{E}]e^{g\circ\pi}))$.

\vspace{0.5cm}

Now we want to examine the complex $\DR~\Psi_{\pi_1}(\pi^{\ast}(\mathcal{M}_{|U})[\ast\widetilde{E}]e^{g\circ\pi})$ locally on $E_d$. Let $T$ be a tubular neighbourhood of $E_d$. According to Corollary \ref{stricttransform} (1), the singular support $\widetilde{S_T}$ of $(\pi^{\ast}(\mathcal{M}_{|U})[\ast\widetilde{E}])_{|T}$ is $\big(E\cup(\cup_{\ell\in\Lambda_{\alpha}}\widetilde{S_\ell})\big)\cap T$.

\vspace{0.5cm}

\noindent
{\bf Computation of the Euler characteristic:} We denote by $Q_1,\ldots,Q_k$ the intersection point of $\cup_{\ell\in\Lambda_{\alpha}}\widetilde{S_\ell}$ with $E_d$.

Let $r$ be the rank of $\mathcal{M}$. As $\pi$ is an isomorphism out of $E$, the rank of $\pi^{\ast}(\mathcal{M}_{|U})$ is $r$ and the multiplicity of the conormal space $T_{\widetilde{S_\ell}}^{\ast}\mathbb{X}$ in the characteristic cycle of $\pi^{\ast}(\mathcal{M}_{|U})$ is $m_\ell$.

\begin{figure}[h]
\includegraphics[height=2.5cm]{euler.eps}
\end{figure}

\begin{enumerate}
\item In a neighbourhood $V\subset\mathbb{X}$ of the point $P$, according to Lemma \ref{resolution}, there exist coordinates $(u,v)$ on $V$ such that $\pi_1(u,v)=uv^n$, $g\circ\pi(u,v)\sim\sfrac{1}{v}$ and the singular support of $\pi^{\ast}(\mathcal{M}_{|U})[\ast\widetilde{E}]$ is included in $uv=0$. Then,
$$\Psi_{\pi_1}(\pi^{\ast}(\mathcal{M}_{|U})[\ast\widetilde{E}]e^{g\circ\pi})_P=\Psi_{uv^n}(\pi^{\ast}(\mathcal{M}_{|U})[\frac{1}{uv}]e^{\frac{1}{v}})_{(0,0)}.$$

Applying Lemma \ref{B} 3., we conclude that the Euler characteristic of the complex $\DR~\Psi_{\pi_1}(\pi^{\ast}(\mathcal{M}_{|U})[\ast\widetilde{E}]e^{g\circ\pi})_P$ is equal to $-r$.

\item In a neighbourhood $V\subset\mathbb{X}$ of a point $Q_i$, $g\circ\pi$ is holomorphic and $\pi^{-1}_1(0)$ is smooth. Then we have
$\DR~\Psi_{\pi_1}(\pi^{\ast}(\mathcal{M}_{|U})[\ast\widetilde{E}]e^{g\circ\pi})_{Q_i}=\DR~\Psi_u(\pi^{\ast}(\mathcal{M}_{|U})[\frac{1}{u}])_{(0,0)}$
and, according to Lemma \ref{A}, the Euler characteristic of the complex $\DR~\Psi_{\pi_1}(\pi^{\ast}(\mathcal{M}_{|U})[\ast\widetilde{E}]e^{g\circ\pi})_{Q_i}$ is $r-\sum_{\widetilde{S_{\ell}}\cap E_d=\{Q_i\}}m_{\ell}$.

\item In a neighbourhood of a point $Q\in E_d\setminus\{P,Q_1,\ldots,Q_k\}$, as in the previous point, we prove that the Euler characteristic of the complex $\DR~\Psi_{\pi_1}(\pi^{\ast}(\mathcal{M}_{|U})[\ast\widetilde{E}]e^{g\circ\pi})_Q$ is $r$ (because $\{\ell\in\Lambda~|~\widetilde{S_\ell}\cap E_d=\{Q\}\}=\emptyset$).
\end{enumerate}

Using the Mayer-Vietoris Theorem, we deduce that the Euler characteristic of the complex $R\Gamma(E_d,\DR~\Psi_{\pi_1}(\pi^{\ast}(\mathcal{M}_{|U})[\ast\widetilde{E}]e^{g\circ\pi}))$ is equal to

\begin{displaymath}
\begin{split}
-r+\sum_{i=1}^k\bigg(r-\sum_{\widetilde{S_{\ell}}\cap E_d=\{Q_i\}}m_{\ell}\bigg)+\chi(E_d\setminus\{P,Q_1,\ldots,Q_k\}).r&=
-\sum_{\widetilde{S_{\ell}}\cap E_d\neq\emptyset}m_{\ell}\\
&=-\sum_{\ell\in\Lambda_{\alpha}}m_{\ell}.
\end{split}
\end{displaymath}

{\bf Computation of the zeta function of the monodromy:} Assume that $(\ast)$ is fulfilled. The fact that for any $\ell\in\Lambda$, $p_\ell=1$ and the assumption $(\ast)$ imply that $\widetilde{S_T}$ is a divisor with normal crossing (cf. Corollary \ref{stricttransform} (2)).

We denote by $T^{\ast}$ the complement of $\widetilde{S_T}$ in $T$ and by $Q_{\ell}$ the intersection point of one $\widetilde{S_\ell}$ with $E_d$, $\ell\in\Lambda_{\alpha}$.

\begin{figure}[h]
\includegraphics[height=2.5cm]{zeta.eps}
\end{figure}

\begin{itemize}
\item[$\bullet$] $(\DR~\pi^{\ast}(\mathcal{M}_{|U})[\ast\widetilde{E}])_{|T^{\ast}}$ is a local system. We can define the monodromy endomorphism around $E_d$, $T_P:(\DR~\pi^{\ast}(\mathcal{M}_{|U})[\ast\widetilde{E}])_{P}\to (\DR~\pi^{\ast}(\mathcal{M}_{|U})[\ast\widetilde{E}])_{P}$, for any $P\in T^{\ast}$. As $T^{\ast}$ is connected and $\widetilde{S_T}$ has normal crossings, the characteristic polynomial of $T_P$ does not depend on $P\in T^{\ast}$. We denote it by $\zeta_r$.
\item[$\bullet$] To any $\ell\in\Lambda_{\alpha}$, we associate the characteristic polynomial $\zeta_\ell$ of the monodromy of the local system $\phi_{\widetilde{h_\ell}}(\DR~\pi^{\ast}(\mathcal{M}_{|U}))_{|\widetilde{S_\ell}\setminus\{(0,0)\}}$ around $(0,0)$, where $\widetilde{h_\ell}$ is an equation of $\widetilde{S_\ell}$. If $h_\ell$ is an equation of $S_\ell$, as $\pi$ is an isomorphism out of $E$, it coincides with the characteristic polynomial of the monodromy of $\phi_{h_\ell}(\DR~\mathcal{M})_{|S_\ell\setminus\{(0,0)\}}$ around $(0,0)$.
\end{itemize}

Using the local behaviour of $\pi_1$ and $g\circ\pi$ on $E_d$ given by Lemma \ref{resolution}, we have:

\begin{enumerate}
\item In a neighbourhood of the point $P$, according to Lemma \ref{B} 3., the zeta function of the monodromy of $\DR~\Psi_{\pi_1}(\pi^{\ast}(\mathcal{M}_{|U})[\ast\widetilde{E}]e^{g\circ\pi})_P$ is equal to $\zeta_r^{-1}$.
\item In a neighbourhood of a point $Q_\ell$, $\ell\in\Lambda_{\alpha}$, according to Lemma \ref{C}, the zeta function of the monodromy of $\DR~\Psi_{\pi_1}(\pi^{\ast}(\mathcal{M}_{|U})[\ast\widetilde{E}]e^{g\circ\pi})_{Q_\ell}$ is equal to $\zeta_r\zeta_\ell^{-1}$.
\item In a neighbourhood of a point $Q\in E_d\setminus\{P,Q_\ell,\ell\in\Lambda_{\alpha}\}$, the zeta function of the monodromy of $\DR~\Psi_{\pi_1}(\pi^{\ast}(\mathcal{M}_{|U})[\ast\widetilde{E}]e^{g\circ\pi})_Q$ is equal to $\zeta_r$.
\end{enumerate}

Then, using the Mayer-Vietoris Theorem, we deduce that the zeta function of the monodromy of $R\Gamma(E_d,\DR~\Psi_{\pi_1}(\pi^{\ast}(\mathcal{M}_{|U})[\ast\widetilde{E}]e^{g\circ\pi}))$ is equal to
$$\zeta_r^{-1}.\prod_{\ell\in\Lambda_{\alpha}}(\zeta_r\zeta_\ell^{-1}).\zeta_r^{\chi(E_d\setminus\{P,Q_\ell,\ell\in\Lambda_{\alpha}\})}=\prod_{\ell\in\Lambda_{\alpha}}\zeta_\ell^{-1}.$$

\section{Realization of  $\mathbb{C}[[t]]\langle\partial_t\rangle$-modules}\label{realization}
Let $\widehat{\mathcal{N}}$ be a formal holonomic $\mathbb{C}[[t]]\langle\partial_t\rangle$-module.

\begin{theorem}\label{realisation}
There exist a ramification $\rho:\tau\to t=\tau^p$ and a regular holonomic $\mathcal{D}_{D\times\mathbb{P}^1}$-module $\mathcal{M}$ such that $\rho^{\ast}(\widehat{\mathcal{N}})$ is isomorphic to the formalization of $\rho^{\ast}(\mathcal{H}^0p_{1+}(\mathcal{M}e^{p_2}))_0$.
\end{theorem}

\begin{proof}
We recall that we identify $\mathcal{D}_{D,0}$ with $\mathbb{C}\{t\}\langle\partial_t\rangle$, by choosing a coordinate $t$ on $D$. We begin by stating two lemmas.

\begin{lemma}\label{BB}
\begin{enumerate}
\item $p_{1+}(\mathcal{O}_{D\times\mathbb{P}^1}e^{p_2})=\mathcal{O}_{D}$.
\item $p_{1+}(\mathcal{O}_{D\times\mathbb{P}^1}[t^{-1}]e^{p_2})=\mathcal{O}_{D}[t^{-1}]$.
\item Let $(p,\alpha)\in\mathbb{N}^{\ast}\times(\tau^{-1}.\mathbb{C}[\tau^{-1}])$. Let $\rho:D^{'}\to D$, defined by $\rho(\tau)=t=\tau^p$. We denote by $Z$ the curve in $D\times\mathbb{P}^1$ parametrized by $\tau\to(\tau^p,\alpha)$ and by $\widetilde{Z}$ the union of $Z$, $\{0\}\times\mathbb{P}^1$ and $D\times\{\infty\}$.
$$\rho^{\ast}p_{1+}(\mathcal{O}_{D\times\mathbb{P}^1}[\ast \widetilde{Z}]e^{p_2})=\underset{\xi^p=1}{\oplus}\mathcal{O}_{D^{'}}[\tau^{-1}]e^{\alpha(\xi\tau)}\oplus \mathcal{O}_{D^{'}}[\tau^{-1}].$$
\end{enumerate}
\end{lemma}

\begin{proof}
We just prove the third point. The first two points can be proved similary.

Let $p_1^{'}:D^{'}\times\mathbb{P}^1\to D^{'}$ and $p_2^{'}:D^{'}\times\mathbb{P}^1\to\mathbb{P}^1$ be the canonical projections and $\rho^{'}=(\rho,id):D^{'}\times\mathbb{P}^1\to D\times\mathbb{P}^1$.

First note that $\rho^{\ast}p_{1+}(\mathcal{O}_{D\times\mathbb{P}^1}[\ast \widetilde{Z}]e^{p_2})=p^{'}_{1+}(\mathcal{O}_{D^{'}\times\mathbb{P}^1}[\ast \rho^{'-1}(\widetilde{Z})]e^{p_2^{'}})$ (cf. Lemma \ref{comm}).

Denote by $\mathcal{K}^{\bullet}:=~^pDR_{D^{'}\times\mathbb{P}^1/D^{'}}(\mathcal{O}_{D^{'}\times\mathbb{P}^1}[\ast \rho^{'-1}(\widetilde{Z})]e^{p_2^{'}})$ the relative de Rham complex of $p_1$ concentrated in negative degrees. It consists in the complex $$\mathcal{O}_{D^{'}\times\mathbb{P}^1}[\ast \rho^{'-1}(\widetilde{Z})]\overset{d}{\to}\mathcal{O}_{D^{'}\times\mathbb{P}^1}[\ast \rho^{'-1}(\widetilde{Z})],$$
with $d(h)=(\partial_{\mathbb{P}^1}(h)+h\partial_{\mathbb{P}^1}(p_2^{'}))$.

By definition, $p_{1+}^{'}(\mathcal{O}_{D^{'}\times\mathbb{P}^1}[\ast \rho^{'-1}(\widetilde{Z})]e^{p_2^{'}})=\mathbb{R}p_{1\ast}^{'}(\mathcal{K}^{\bullet})$. We consider the spectral sequence $\mathcal{H}^{i-k}(R^kp_{1\ast}^{'}(\mathcal{K}^{\bullet}),d)\Longrightarrow \mathbb{R}^ip_{1\ast}^{'}(\mathcal{K}^{\bullet})$, where $(R^kp_{1\ast}^{'}(\mathcal{K}^{\bullet}),d)$ denotes the complex $R^kp_{1\ast}^{'}(\mathcal{O}_{D^{'}\times\mathbb{P}^1}[\ast \rho^{'-1}(\widetilde{Z})])\overset{d}{\to}R^kp_{1\ast}^{'}(\mathcal{O}_{D^{'}\times\mathbb{P}^1}[\ast \rho^{'-1}(\widetilde{Z})])$.

We claim that $R^kp_{1\ast}^{'}(\mathcal{O}_{D^{'}\times\mathbb{P}^1}[\ast \rho^{'-1}(\widetilde{Z})])=0$, for $k\neq 0$. Indeed, let $j:(D^{'}\times\mathbb{P}^1)\setminus\rho^{'-1}(\widetilde{Z})\hookrightarrow D^{'}\times\mathbb{P}^1$ and $P\in D^{'}$:
$$\begin{array}{ll}
R^kp_{1\ast}^{'}&(\mathcal{O}_{D^{'}\times\mathbb{P}^1}[\ast \rho^{'-1}(\widetilde{Z})])_P=\\
&=R^kp_{1\ast}^{'}Rj_{\ast}j^{-1}(\mathcal{O}_{D^{'}\times\mathbb{P}^1})_P,\\
&=R^k(p_1^{'}\circ j)_{\ast}j^{-1}(\mathcal{O}_{D^{'}\times\mathbb{P}^1})_P,\\
&=\varinjlim_{U\ni P open}R\Gamma(j^{-1}(U\times\mathbb{P}^1),j^{-1}(\mathcal{O}_{D^{'}\times\mathbb{P}^1})).\\
\end{array}$$

As $j^{-1}(U\times\mathbb{P}^1)=(U\times\mathbb{C})\setminus\rho^{'-1}(Z\cup\{0\}\times\mathbb{C})$ is Stein, we conclude that  $R^kp_{1\ast}^{'}(\mathcal{O}_{D^{'}\times\mathbb{P}^1}[\ast \rho^{'-1}(\widetilde{Z})])=0$, for $k\neq 0$.

Now $(R^0p_{1\ast}^{'}(\mathcal{K}^{\bullet}),d)$ is the complex:
$$\mathcal{O}_{D^{'}}[y]\bigg[\frac{1}{\tau\underset{\xi^p=1}{\prod}(y-\alpha(\xi\tau))}\bigg]\overset{d}{\to} \mathcal{O}_{D^{'}}[y]\bigg[\frac{1}{\tau\underset{\xi^p=1}{\prod}(y-\alpha(\xi\tau))}\bigg],$$
with $d(h)=(\partial_y(h)+h)$.

\begin{itemize}
\item Computation of $\mathcal{H}^{-1}(R^0p_{1\ast}^{'}(\mathcal{K}^{\bullet}),d)$:\\
Let $h\in\mathcal{O}_{D^{'}}[y]\bigg[\frac{1}{\tau\underset{\xi^p=1}{\prod}(y-\alpha(\xi\tau))}\bigg]$ such that $d(h)=0$. Thus $h=Ce^{-y}$, with $C\in\mathcal{O}_{D^{'}}[\frac{1}{\tau}]$ and then $h=0$. Thus, $\mathcal{H}^{-1}(R^0p_{1\ast}^{'}(\mathcal{K}^{\bullet}),d)=0$.
\item Computation of $\mathcal{H}:=\mathcal{H}^{0}(R^0p_{1\ast}^{'}(\mathcal{K}^{\bullet}),d)$:\\
We note first that $\mathcal{H}$ is generated over $\mathcal{O}_{D^{'}}[\frac{1}{\tau}]$ by \\$\{y^m,\frac{y^m}{(y-\alpha(\xi\tau))^n};~m\geq 0,~n\geq 1 \text{ and }\xi^p=1\}$.

Moreover, if $m,n>0$, we have:
$\frac{y^m}{(y-\alpha(\xi\tau))^n}=\frac{y^{m-1}}{(y-\alpha(\xi\tau))^{n-1}}+\frac{y^{m-1}\alpha(\xi\tau)}{(y-\alpha(\xi\tau))^n}$ and $y^m\equiv -my^{m-1}$.

Then $\mathcal{H}$ is generated over $\mathcal{O}_{D^{'}}[\frac{1}{\tau}]$ by $\{1,\frac{1}{(y-\alpha(\xi\tau))^n};~n\geq 1 \text{ and }\xi^p=1\}$. But we have $\frac{1}{(y-\alpha(\xi\tau))^n}\equiv\frac{n}{(y-\alpha(\xi\tau))^{n+1}}$. Then, $\mathcal{H}$ is generated over $\mathcal{O}_{D^{'}}[\frac{1}{\tau}]$ by $\{1,\frac{1}{(y-\alpha(\xi\tau))};~\xi^p=1\}$.

Then we have  $\mathcal{H}=\underset{\xi^p=1}{\oplus}\mathcal{O}_{D^{'}}[\tau^{-1}].\frac{1}{(y-\alpha(\xi\tau))}\oplus \mathcal{O}_{D^{'}}[\tau^{-1}]$.

As $
\partial_{\tau}(\frac{1}{y-\alpha(\xi\tau)})=\partial_{\tau}(\alpha(\xi\tau)).\frac{1}{(y-\alpha(\xi\tau))^2}\equiv\partial_{\tau}(\alpha(\xi\tau)).\frac{1}{y-\alpha(\xi\tau)}$, we have an isomorphism of $\mathcal{D}_{D^{'}}$-modules
$$\mathcal{O}_{D^{'}}[\tau^{-1}].\frac{1}{(y-\alpha(\xi\tau))}=\mathcal{O}_{D^{'}}[\tau^{-1}]e^{\alpha(\xi\tau)}.$$

Lemma \ref{BB} follows.
\end{itemize}

\end{proof}

\begin{lemma}\label{AA}
Let $\mathcal{N}$ be a holonomic $\mathcal{D}_{D}$-module and $\mathcal{M}$ be a holonomic $\mathcal{D}_{D\times\mathbb{P}^1}$-module. Suppose that $\mathcal{M}$ is $\mathcal{O}_{D\times\mathbb{P}^1}$-flat. We have an isomorphism: $$p_{1+}(\mathcal{M}\otimes_{\mathcal{O}_{D\times\mathbb{P}^1}}p_1^{\ast}(\mathcal{N}))=p_{1+}(\mathcal{M})\otimes_{\mathcal{O}_{D}}^{\mathbb{L}}\mathcal{N}.$$
\end{lemma}

\begin{proof}
First, we remark that $\mathcal{M}\otimes_{\mathcal{O}_{D\times\mathbb{P}^1}}p_1^{\ast}\mathcal{N}=\mathcal{M}\otimes_{p_1^{-1}\mathcal{O}_D}p_1^{-1}\mathcal{N}$, with the structure of $\mathcal{D}_{D\times\mathbb{P}^1}$-module given by $\xi(m\otimes n)=\xi(m)\otimes n+\xi(p_1)m\otimes \partial_t(n)$, where $\xi$ is a local section of the sheaf of vector fields on $D\times\mathbb{P}^1$.

Denote by $~^p\DR_{D\times\mathbb{P}^1/D}$ (resp. $\Omega^{\bullet+1}_{D\times\mathbb{P}^1/D}$) the relative de Rham functor of $p_1$ (resp. the complex of relative forms of $p_1$). The differential $d$ of the complex $~^p\DR_{D\times\mathbb{P}^1/D}(\mathcal{M}\otimes_{p_1^{-1}\mathcal{O}_D}p_1^{-1}\mathcal{N})=\Omega^{\bullet+1}_{D\times\mathbb{P}^1/D}\otimes_{\mathcal{O}_{D\times\mathbb{P}^1}}(\mathcal{M}\otimes_{p_1^{-1}\mathcal{O}_D}p_1^{-1}\mathcal{N})$ is defined by:
$$\begin{array}{lll}
d(w\otimes(m\otimes n))&=&dw\otimes(m\otimes n)+dp_2\wedge w\otimes\partial_{\mathbb{P}^1}(m\otimes n),\\
&=&dw\otimes(m\otimes n)+dp_2\wedge w\otimes(\partial_{\mathbb{P}^1}(m)\otimes n),
\end{array}$$
and the action of $p_1^{-1}\mathcal{D}_D$ is given by:
$$\partial_t(w\otimes(m\otimes n))=w\otimes(\partial_tm\otimes n)+w\otimes(m\otimes\partial_tn).$$

Then, we have an isomorphism of $p_1^{-1}\mathcal{D}_D$-modules:
$$
~^p\DR_{D\times\mathbb{P}^1/D}(\mathcal{M}\otimes_{p_1^{-1}\mathcal{O}_D}p_1^{-1}\mathcal{N})=~^p\DR_{D\times\mathbb{P}^1/D}(\mathcal{M})\otimes_{p_1^{-1}\mathcal{O}_D}p_1^{-1}\mathcal{N}.$$

We conclude with:
$$\begin{array}{ll}
p_{1+}(\mathcal{M}&\otimes^{\mathbb{L}}_{\mathcal{O}_{D\times\mathbb{P}^1}}p_1^{\ast}(\mathcal{N}))=\\
&=\mathbb{R}p_{1\ast}(~^p\DR_{D\times\mathbb{P}^{1}/D}(\mathcal{M}\otimes_{\mathcal{O}_{D\times\mathbb{P}^1}}p_1^{\ast}(\mathcal{N}))),\\
&=\mathbb{R}p_{1\ast}(~^p\DR_{D\times\mathbb{P}^{1}/D}(\mathcal{M})\otimes_{p_1^{-1}\mathcal{O}_D}p_1^{-1}(\mathcal{N})),\\
&=\mathbb{R}p_{1\ast}(~^p\DR_{D\times\mathbb{P}^{1}/D}(\mathcal{M})\otimes^{\mathbb{L}}_{p_1^{-1}\mathcal{O}_D}p_1^{-1}(\mathcal{N})),\\
&(~^p\DR_{D\times\mathbb{P}^{1}/D}(\mathcal{M})\text{ is }p_1^{-1}\mathcal{O}_D\text{-flat),}\\
&=\mathbb{R}p_{1\ast}(~^p\DR_{D\times\mathbb{P}^{1}/D}(\mathcal{M})\otimes^{\mathbb{L}}_{p_1^{-1}\mathcal{D}_D}(p_1^{-1}\mathcal{D}_D\otimes_{p_1^{-1}\mathcal{O}_D}p_1^{-1}(\mathcal{N}))),\\
&=\mathbb{R}p_{1\ast}(~^p\DR_{D\times\mathbb{P}^{1}/D}(\mathcal{M}))\otimes^{\mathbb{L}}_{\mathcal{D}_D}(\mathcal{D}_D\otimes_{\mathcal{O}_D}\mathcal{N}),\\
&\text{(Proposition 2.6.6 in \cite{KaSc}),}\\
&p_{1+}(\mathcal{M})\otimes^{\mathbb{L}}_{\mathcal{O}_D}\mathcal{N}.
\end{array}$$
\end{proof}

According to the structure of a formal $\mathbb{C}[[t]]\langle\partial_t\rangle$-module (cf. \cite{Ma}), we are led to prove the theorem in the cases of regular modules and purely irregular modules.
\begin{enumerate}
\item $\underline{\text{Regular case:}}$ Suppose $\widehat{\mathcal{N}}$ regular.
There exist a small disc $D$ in $\mathbb{C}$ centered at the origin and a regular holonomic $\mathcal{D}_D$-module $\mathcal{N}$, such that the formalization of $\mathcal{N}_0$ is $\widehat{\mathcal{N}}$ (cf. Theorem 5.3 p. 38 in \cite{Ma}).
Let $\mathcal{M}$ be the regular holonomic $\mathcal{D}_{D\times\mathbb{P}^1}$-module $p_1^{\ast}(\mathcal{N})$. We have:
$$\begin{array}{ll}
p_{1+}(\mathcal{M}e^{p_2})&=p_{1+}(\mathcal{O}_{D\times\mathbb{P}^1}e^{p_2}\otimes_{\mathcal{O}_{D\times\mathbb{P}^1}}p_1^{\ast}(\mathcal{N})),\\
&=p_{1+}(\mathcal{O}_{D\times\mathbb{P}^1}e^{p_2})\otimes_{\mathcal{O}_D}^{\mathbb{L}}\mathcal{N},\text{ (Lemma \ref{AA}),}\\
&=\mathcal{O}_D\otimes_{\mathcal{O}_D}^{\mathbb{L}}\mathcal{N},\text{ (Lemma \ref{BB}),}\\
&=\mathcal{N}.
\end{array}$$

\item $\underline{\text{Purely irregular case:}}$ Let $(p,\alpha)\in \mathbb{N}^{\ast}\times(\tau^{-1}.\mathbb{C}[\tau^{-1}])$ and $\widehat{R_{\alpha}}$ be a regular holonomic $\mathbb{C}[[\tau]]\langle\partial_{\tau}\rangle$-module, such that $\widehat{R_{\alpha}}[\tau^{-1}]=\widehat{R_{\alpha}}$. Let $\rho:\tau\to t=\tau^p$ and suppose $\rho^{\ast}(\widehat{\mathcal{N}})=\underset{\xi^p=1}{\oplus}\widehat{R_{\alpha}}e^{\alpha(\xi\tau)}$.

There exist a small disc $D^{'}$ in $\mathbb{C}$ centered at the origin and a regular holonomic $\mathcal{D}_{D^{'}}$-module $\widetilde{R_{\alpha}}$ such that the formalization of $(\widetilde{R_{\alpha}})_0$ is $\widehat{R_{\alpha}}$ (cf. Theorem 5.3 p. 38 in \cite{Ma}).But there exists a regular holonomic $\mathcal{D}_{D}$-module $R_{\alpha}$ such that $\widetilde{R_{\alpha}}=\rho^{\ast}(R_{\alpha})$, where $\rho:D^{'}\to D$, $\rho(\tau)=t=\tau^p$. Indeed, as $\rho^{\ast}$ is exact and $\widetilde{R_{\alpha}}$ is regular holonomic with $\widetilde{R_{\alpha}}[\tau^{-1}]=\widetilde{R_{\alpha}}$, we can reduce the proof to the case where
$\widetilde{R_{\alpha}}=\sfrac{\mathcal{D}_{D^{'}}}{\mathcal{D}_{D^{'}}((\tau\partial_{\tau}-\alpha)^r)}$, with $r\in\mathbb{N}^{\ast}$ and $\alpha\in\mathbb{C}\setminus(-\mathbb{N})$ (cf. Corollary 19 in \cite{BrMai}). Then, we consider $R_{\alpha}=\sfrac{\mathcal{D}_D}{\mathcal{D}_D((t\partial_t-\frac{\alpha}{p})^r)}$.

Using notations of Lemma \ref{BB}, we denote by $\mathcal{M}$ the regular holonomic  $\mathcal{D}_{D\times\mathbb{P}^1}$-module $\sfrac{p_1^{\ast}(R_{\alpha})[\ast \widetilde{Z}]}{p_1^{\ast}(R_{\alpha})[t^{-1}]}$.
According to Lemmas \ref{AA} and \ref{BB}, we have two isomorphisms:
$$\begin{array}{l}
\rho^{\ast}p_{1+}(p_1^{\ast}(R_{\alpha})[\ast \widetilde{Z}]e^{p_2})=\underset{\xi^p=1}{\oplus}\widetilde{R_{\alpha}}e^{\alpha(\xi\tau)}\oplus \widetilde{R_{\alpha}}\\
\rho^{\ast}p_{1+}(p_1^{\ast}(R_{\alpha})[t^{-1}]e^{p_2})=\widetilde{R_{\alpha}}.
\end{array}$$

Then the formalization of $\rho^{\ast}p_{1+}(\mathcal{M}e^{p_2})$ is $\underset{\xi^p=1}{\oplus}\widehat{R_{\alpha}}e^{\alpha(\xi\tau)}$.
\end{enumerate}

\end{proof}

\section{Direct image of holonomic $\mathcal{D}$-module of exponential type: the algebraic case}\label{algebraic}

In this section, we prove Theorem \ref{2}.
We consider $\mathcal{N}^k:=\mathcal{H}^kj_+f_+(\mathcal{M}e^g)$, the extension of $\mathcal{H}^kf_+(\mathcal{M}e^g)$ at infinity, where $j:\mathbb{A}^1\hookrightarrow\mathbb{A}^1\cup\{\infty\}\simeq\mathbb{P}^1$. Let $c\in\mathbb{P}^1$. The proof will be divided in two steps.
\begin{enumerate}
\item We consider the following diagrams
$$\xymatrix{\mathbb{C}^2\ar@{^(->}[r]^-i\ar[d]^-{\pi_1}&\mathbb{P}^1\times\mathbb{P}^1\ar[d]^-{p_1}&B_c\times\mathbb{P}^1\ar@{_(->}[l]_-{i_c}\ar[d]^{p_1}
&\mathbb{C}^2\ar@{^(->}[r]^-i\ar[d]^-{\pi_2}&\mathbb{P}^1\times\mathbb{P}^1\ar[d]^-{p_2}&B_c\times\mathbb{P}^1\ar@{_(->}[l]_-{i_c}\ar[d]^{p_2}\\
\mathbb{C}\ar@{^(->}[r]^-j&\mathbb{P}^1&B_c\ar@{_(->}[l]_-{j_c},&\mathbb{C}\ar@{^(->}[r]^-j&\mathbb{P}^1&\mathbb{P}^1\ar@{=}[l],}$$
where $\pi_1$, $\pi_2$ (resp. $p_1$, $p_2$) are the two canonical
projections.

Then we have:
$$\begin{array}{lll}
j_c^{\ast}(j_+f_+(\mathcal{M}e^g))^{\mathrm{an}}&=&j_c^{\ast}(j_+\pi_{1+}(f,g)_+(\mathcal{M}e^g))^{\mathrm{an}},\\
&&\text{(Prop. 6.4 in \cite{Bo} and }f=\pi_1\circ(f,g)),\\
&=&j_c^{\ast}(p_{1+}i_+(f,g)_+(\mathcal{M}e^g))^{\mathrm{an}},\\
&&\text{(Prop. 6.4 in \cite{Bo} and }j\circ\pi_1=p_1\circ i),\\
&=&j_c^{\ast}p_{1+}(i_+(f,g)_+(\mathcal{M}e^g))^{\mathrm{an}},\\
&&\text{(Prop 8.2.2 p. 179 in \cite{Me})},\\
&=&p_{1+}i_c^{\ast}((i_+(f,g)_+(\mathcal{M}))e^{p_2})^{\mathrm{an}},\\
&&(i_c\text{ is an open inclusion}),\\
&=&p_{1+}(\mathcal{P}_ce^{p_2}).
\end{array}$$

As the formalization of the germ $\mathcal{N}_c^k$ and of its analytization are equal, $\mathcal{N}_c^k$ and $\mathcal{H}^kp_{1+}(\mathcal{P}_ce^{p_2})_c$ have the same formal irregular part.

\item Then we claim that $\mathcal{H}^kp_{1+}(\mathcal{P}_ce^{p_2})_c$ and $\mathcal{H}^0p_{1+}(\mathcal{H}^k\mathcal{P}_ce^{p_2})_c$ have the same formal irregular part. According to the formal decomposition Theorem (cf. Theorem 1.2 p. 43 and Theorem 2.3 p. 51 in \cite{Ma}), there exists a ramification $\rho:D^{'}_c\to D_c$, $\rho(\tau)=\tau^p=t$, such that the modules $\rho^{\ast}(\mathcal{H}^kp_{1+}(\mathcal{P}_ce^{p_2}))_c$ and $\rho^{\ast}(\mathcal{H}^0p_{1+}(\mathcal{H}^k\mathcal{P}_ce^{p_2}))_c$ are isomorphic to the finite direct sum of some formal modules of exponential type. Then, it is sufficient to prove that, given an element $\alpha\in\tau^{-1}.\mathbb{C}[\tau^{-1}]$, the regular part of  $(\rho^{\ast}(\mathcal{H}^kp_{1+}(\mathcal{P}_ce^{p_2}))_c)e^{-\alpha}$ and  $(\rho^{\ast}(\mathcal{H}^0p_{1+}(\mathcal{H}^k\mathcal{P}_ce^{p_2}))_c)e^{-\alpha}$ are isomorphic. Thus, we are led to prove that the specialization at $c$ of  $(\rho^{\ast}(\mathcal{H}^kp_{1+}(\mathcal{P}_ce^{p_2})))e^{-\alpha}$ and  $(\rho^{\ast}(\mathcal{H}^0p_{1+}(\mathcal{H}^k\mathcal{P}_ce^{p_2})))e^{-\alpha}$ are isomorphic (cf. Example 5.2.1 in \cite{LaMa}).

Let $\rho^{'}=(\rho,id):D^{'}_c\times\mathbb{P}^1\to  D_c\times\mathbb{P}^1$. We denote by $p_1^{'}:D_c\times\mathbb{P}^1\to D_c$ and $p_2^{'}:D_c\times\mathbb{P}^1\to\mathbb{P}^1$ the canonical projection. Let $\overline{p}_1^{'}:\{c\}\times\mathbb{P}^1\to\{c\}$. We have:
$$\begin{array}{ll}
\spe_c((\rho^{\ast}&(\mathcal{H}^kp_{1+}(\mathcal{P}_ce^{p_2})))e^{-\alpha})_c\\
&=\spe_c(\mathcal{H}^kp_{1+}^{'}(\rho^{'\ast}(\mathcal{P}_c)e^{p_2^{'}})e^{-\alpha})_c,\\
&\text{(Lemma \ref{comm} extended to complexes of $\mathcal{D}$-modules),}\\
&=\spe_c(\mathcal{H}^kp_{1+}^{'}(\rho^{'\ast}(\mathcal{P}_c)e^{p_2^{'}-\alpha\circ p_1^{'}}))_c,\\
&=\mathcal{H}^k\overline{p}_{1+}^{'}\spe_{\{c\}\times\mathbb{P}^1}(\rho^{'\ast}(\mathcal{P}_c)e^{p_2^{'}-\alpha\circ p_1^{'}})_c,\\
&\text{(Theorem 9.4.1 in \cite{LaMa}),}\\
&=R^k\Gamma(\{c\}\times\mathbb{P}^1,\DR~\spe_{\{c\}\times\mathbb{P}^1}(\rho^{'\ast}(\mathcal{P}_c)e^{p_2^{'}-\alpha\circ p_1^{'}})[+1]),\\
&\text{(p.5 in \cite{Ma3}).}
\end{array}$$

According to Proposition \ref{11} extended to complexes of $\mathcal{D}$-modules, the support of  $\spe_{\{c\}\times\mathbb{P}^1}(\rho^{'\ast}(\mathcal{P}_c)e^{p_2^{'}-\alpha\circ p_1^{'}})$ is included in $(c,\infty)$. Then,

$$\begin{array}{ll}
\spe_c&((\rho^{\ast}(\mathcal{H}^kp_{1+}(\mathcal{P}_ce^{p_2})))e^{-\alpha})_c=\\
&=(\mathcal{H}^k\DR~\spe_{\{c\}\times\mathbb{P}^1}(\rho^{'\ast}(\mathcal{P}_c)e^{p_2^{'}-\alpha\circ p_1^{'}})[+1])_{(c,\infty)},\\
&=(\DR~\mathcal{H}^k\spe_{\{c\}\times\mathbb{P}^1}(\rho^{'\ast}(\mathcal{P}_c)e^{p_2^{'}-\alpha\circ p_1^{'}})[+1])_{(c,\infty)},\\
&=R^0\Gamma(\{c\}\times\mathbb{P}^1,\DR~\mathcal{H}^k\spe_{\{c\}\times\mathbb{P}^1}(\rho^{'\ast}(\mathcal{P}_c)e^{p_2^{'}-\alpha\circ p_1^{'}})[+1]),\\
&=R^0\Gamma(\{c\}\times\mathbb{P}^1,\DR~\spe_{\{c\}\times\mathbb{P}^1}(\rho^{'\ast}(\mathcal{H}^k\mathcal{P}_c)e^{p_2^{'}-\alpha\circ p_1^{'}})[+1]),\\
&=\mathcal{H}^0\overline{p}_{1+}^{'}\spe_{\{c\}\times\mathbb{P}^1}(\rho^{'\ast}(\mathcal{H}^k\mathcal{P}_c)e^{p_2^{'}-\alpha\circ p_1^{'}})_c,\\
&\text{(p.5 in \cite{Ma3}),}\\
&=\spe_c(\mathcal{H}^0p_{1+}^{'}(\rho^{'\ast}(\mathcal{H}^k\mathcal{P}_c)e^{p_2^{'}-\alpha\circ p_1^{'}}))_c,\\
&\text{(Theorem 4.8.1 p. 226 in \cite{MeSa}),}\\
&=\spe_c(\mathcal{H}^0p_{1+}^{'}(\rho^{'\ast}(\mathcal{H}^k\mathcal{P}_c)e^{p_2^{'}})e^{-\alpha})_c,\\
&=\spe_c((\rho^{\ast}(\mathcal{H}^0p_{1+}(\mathcal{H}^k\mathcal{P}_ce^{p_2})))e^{-\alpha})_c,\\
&\text{(Lemma \ref{comm}).}
\end{array}$$
\end{enumerate}

redpajama_set_name: RedPajamaArXiv


text:
Q: How to remove registration information of VS 2010 from a computer without reinstall Windows I want to remove visual studio 2010 from a computer i am selling but even after uninstall, it looks like my product key is still around - if you re-install it it will use the product key...
How do I remove that registration info from the computer as I don't want the new owner of the computer to use my product key. And I don't want to have to uninstall Windows.

A: Search the registry and delete everything that refers to Visual Studio and you should clobber the stored key amongst other things.
I would recommend wiping the drive (just zeroing it would probably suffice, a full secure wipe if feeling unnecessarily paranoid) and reinstalling anyway though - who knows what data you might accidentally leave lying around (or easily undeleted) otherwise, like usernames and passwords to online accounts you care about or if you have ever stored banking details on there (your buyer may not abuse any such information if found, but if they get infected by malware that goes searching for such data your accounts could be compromised).


Afghan asylum seeker Riaz Khan Ahmadzai (17) attempts to kill several people on an regional train with an ax and a knife. SEK commando shoots the terrorist. He confessed to the Islamic State. He lived with a German foster family. Main victim was a Chinese family who were traveling as tourists. A 27-year-old woman no longer walks without a cap because of the serious injuries. The father, an engineer and main provider of the family, was still unable to work in November 2016. His brain functions are not yet fully restored, which causes him to concentrate hard and react slowly. The 17-year-old, uninjured son is still under psychiatric treatment.
A Moroccan (26) is arrested by special forces of the LKA in a refugee home near Leipzig. Im is accused of preparing an assassination in Berlin - for the Russian Embassy. Motivation seems to be the Russian support of Assad's government in Syria. Already on February 8, the man had triggered a terror alert. At that time he had entered the local education and technology center with a backpack and announced a bomb attack to students. More than 100 policemen were on duty - fortunately an empty threat. The offender, who had then set himself, was released.
Charge against two Syrians, Kamel T.H.J. (24) and Azad R. (22), on suspicion of involvement in combat operations for the foreign terrorist group "Ahrar al-Sham".
The Syrian Ismail A. J. (38) is arrested! He is strongly suspected of having supported the foreign terrorist group â € œJabhat al-Nusraâ €. He was a member of the militia "Liwa Muâ € ™ ta" in the Syrian Civil War, which cooperated with and supported the goals of the terrorist group â € œJabhat al-Nusraâ € throughout 2013. In the fight for a military airport, he took an active part in the â € œLiwa Muâ € taâ €. Custody!
The German Dasbar W, (29), son of Iraqi parents, is arrested shortly before the impending attack. He wanted to rouse the car park in Karlsruhe with a car, according to the investigators, to kill as many people as possible. His network reaches to the IS in Syria, where he was also active.
The Syrian asylum seeker Mohammad Y. (23) stabs a man unknown to him, who, accompanied by his wife, walks across St. Kassians Square, without cause, with a knife in the back from behind, wants to flee, but is held by passers-by , The victim was in mortal danger. The asylum seeker lives since 2013 in Germany. "So far, there was no evidence of a politically motivated act, but rather preliminary investigations point to a mental illness of the 23-year-old," is found immediately after the fact. During the trial from January 2018, the offender says â € œthat a voice speaks to himâ € and that â € œGod ordered him to do so, â € say police officers who are summoned as witnesses in court. Because this "God" Allah is an arbitrary Islamic terror. The victim suffers, according to his own statement, even today under the assassination attempt: he has problems breathing when he talks a long time, the knife stab has hurt his lungs. He was ill for 3 months after the attempted murder, still has chest pains today, suffers from concentration problems and anxiety attacks and is undergoing psychological treatment.
Police-savvy Libyan Ayman S. (28) suddenly attacks a 56-year-old woman and stabbed her down with a kitchen knife for disturbing her summery clothes. The woman is hospitalized with serious injuries. In another eight cases, the offender has attacked and punched people with a chain, dog leash or metal bar. A psychiatrist determines his indebtedness. Start of the trial!
A Syrian-suspected Syrian is arrested by a special task force of the NRW police. The man is suspected of being a member of a foreign terrorist group. At the same time, searches are being conducted in Hesse, Saxony and Lower Saxony.


text:
Home » Holiday guides » Scotland: Skye locals' sentiment stacks up against Fairy Glen tourist stone pile cairns
Scotland: Skye locals' sentiment stacks up against Fairy Glen tourist stone pile cairns
2018-09-21 mediabest Holiday guides
Tourists visiting beauty spots in the Scottish Highlands are putting them at risk of erosion by building rock formations for Instagram snaps, conservationists have said.
Stacked pyramids of stones have appeared on Unesco-listed beaches on Orkney, and at the Fairy Glen on the Isle of Skye – made famous by blockbuster film The BFG.
Stone stacking, which enthusiasts describe as "meditative" and "artistic", can also be seen on Iona, Inner Hebrides. However, one conservationist said "without a doubt it is having an impact on ecology".
Dog-walker Claire Irons, 53, spotted at least 50 of the rock formations at the Fairy Glen.
The area is known as the "Fairy Glen" due to a distinctive group of round grassy hills, with ponds between them.
Concerned the rocks might topple and crush her Border Terrier, Treacle, Mrs Irons confronted tourists she saw stacking rocks and told them to stop – but was told she was spoiling their fun.
Mrs Irons said: "It's the first time I've noticed it, we don't tend to go up that way in summer. There's normally so many tourists around. When I went up on a dog walk, I was amazed – you couldn't walk on the path.
"It is tourists rather than locals who do it."
Having lived on the island for five years, Mrs Irons said she was concerned that the structure of stone walls was being destroyed by tour groups.
They have also been spotted on Iona.
Mr Hourston added: "The people who like to do this see it as completely harmless. People are increasingly wanting to get into the wild, but the wild is becoming less wild."
Issues with tourists hit the Isle of Skye last year, too.
It waved the white flag after it was overwhelmed with visitors.
So many tried to visit that the police warned them not to come if they hadn't booked accommodation.
Hotels and campsites were so full that they turned people away.
Some holidaymakers slept in their vehicles while the sheer number of cars, coaches and camper vans created bottlenecks on the narrow roads.
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