text:
Article from BTNews 10 MAY 2010
Mercedes-Benz C180 CGI Sport Blue Efficiency
Ultra Modern and Prestigious Too!
Pioneering, class leading, best in class, bench mark…I hear all these superlatives during numerous new car launches as manufacturers strive to convince journalists and potential buyers that their latest product is stronger in every aspect, including brand image, than the earlier models.
Such is the pace of technological developments, driven on not only to garner greater sales but by a quest to save the planet and save the lives of all who have anything to do with motor cars, that a new car does not remain new for long, the quest to remain ahead of the game is eternal.
In among all this automotive activity is an obvious intent to create product creditability and there is no better exponent than Mercedes-Benz, who has been at practicing the art for over 100 years.
These thoughts went through my mind when testing the latest Mercedes-Benz C-Class Estate that, in company with the saloon version, has remained a top seller in the company's extensive model array for donkey's ages.
Clearly Mercedes-Benz has an established pedigree and continues to maintain the status by continually improving on all aspects of its activities without being unduly upbeat.
What is evident is that this model (C-class goes back to 1993) has grown in stature, for size proving to be a more than acceptable 5-seat compact executive car and for the ever increasing range of variants on offer.
Opt for a C-Class and then work out whether you want a more than competent small capacity diesel or petrol engine, something in between or a super car version delivering 272 bhp and bristling with performance enhancing technology.
The common denominator is a finely honed platform, body shell, suspension and running gear that has an abundance of ability to complement any chosen power unit. Here is an effortless form of driving satisfaction that is synonymous with this company's products.
I understood Mercedes' reasoning in providing a petrol powered test car rather than the commonly accepted turbo diesel engine when applied to an estate car as the exercise proved that great advances have been achieved in the efficiency of the petrol unit, this car being an example of the company's Blue Efficiency technology, a programme embracing both types of popular fuel, hybrid technology, electric power and stop+start systems.
Thus this smallest capacity 4-cylinder 1.8 litre Blue Efficiency engine delivers a highly competitive 156 bhp at 5,000 rpm in a manner that suggest a much larger power unit with at least six cylinders.
Complement this delightful engine with a state of the art 5-speed automatic transmission and the result proved to be a more than satisfying turn of performance with a claimed top speed of 132 mph, more than brisk acceleration to 62 mph in 9.2 seconds and, wait for it, a combined fuel consumption of 38.2 mpg.
Mercedes-Benz has never believed in feeble performing cars and with this test car the performance is all the more creditable when appreciating this an executive class estate car capable of carrying five, at least average sized, adults and as much or possible more luggage than any similar sized estate car.
This prestigious though far from overbearing Mercedes-Benz model provides both a driving character that masterfully combines mechanical refinement with impressive ride and handling qualities plus comfort levels that are hard to beat, especially the driver's 'workstation' that employs both powered and manual seat adjust systems to tailor it to a fine fit.
Of course, there are some established Mercedes-Benz features that have been familiar for years, including the foot operated parking brake and the single stalk for operating both windscreen wipers and headlamp dipping that caused strong comment not so many years ago but have clearly been accepted/appreciated by droves of discerning owners.
I miss not seeing the familiar three-pointed star mascot on top of the bonnet (health and safety, you know!) as it gave a certain air of breeding though cast an eye around the interior and admire the appointments, the quality of material, the tasteful yet logical design and I suggest this is undoubtedly close to a work of art.
For equipment, driver aids, safety aids, lifestyle enhancement, this SE version has a breathtakingly extensive equipment list. I can fully understand why so many owners or potential owners make the pilgrimage to Mercedes-Benz World situated on the old Brooklands race track in Weybridge, Surrey to find out what the car is all about – there's no new car demonstration quite like it! www.mercedes-benz.co.uk/world
This C-Class Mercedes-Benz estate is a worthy example of a car brand that has kept pace with a changing world, combining a priceless reputation with down to earth involvement to provide an extensive and desirable car range. Easy to understand why customer loyalty is strong, as one owner said to me: "Get bitten by the Mercedes-Benz bug , and it never lets go."
Rivals include: BMW 3 Series Touring, Alfa
Romeo 159 Estate, Volvo V70,Audi A4 Avant.
STAR RATINGS (out of 10)
Price from: £26,215 on the road – as tested £32,200
NOTES FROM TED WILKINSON'S MOTORING DIARY
VOLKSWAGEN: An insight into future economy models from Volkswagen has revealed that a Golf Blue-e-motion electric model will go into production in 2011, powered by a 110 bhp electric motor good for 86 mph and a range of 90 miles. Between 2012 and 2013 a range of hybrid powered models will be available, including the Touareg 4x4, Jetta saloon, Golf and Passat.
AUDI: An ultra potent version of the Audi R8 has been revealed. Powered by a 553 bhp V10 engine it is claimed to be the fastest production Audi at 199 mph. UK allocation is limited the 33 cars, priced at £142,585 on the road.
TRACKER: Motor cycle thieves were caught out when they attempted to steal a £10,000 Suzuki motorcycle from an office parking space in London. Not only caught on CCTV when loading the machine into their van they were then apprehended by police when the TRACKER device activated.
MITSUBISHI: A new compact cross-over model, the Mitsubishi ASX has been launched. Competing in the C-segment against VW Golf and Ford Focus the new model is powered by two high efficiency Euro 5 compatible engines, a 116 bhp
1.6 petrol and a 147 bhp 1.8 DiD turbo diesel. The latter unit is claimed to be the first diesel engine to feature variable valve technology – both 2 and 4-W-D versions are available.
OUR READERS' FINEST WORDS (All times and dates are GMT)
All comments are filtered to exclude any excesses but the Editor does not have to agree with what is being said. 100 words maximum
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Anyone can zip through the lanes of an open highway in an illegal street race--the real challenge is driving the right way! Killing, crashing, and driving the wrong way won't get you too far. The traffic laws are tight in this part of town. No driving through red lights, either--not even to turn right! Pick from a variety of cars to navigate your way through the law-abiding world of driving.


text:
BVWire Issue #151-3 | April 15, 2015
Damodaran questions the small-cap premium
"I have never used a small cap premium when valuing a company and I don't plan to start now," states Professor Aswath Damodaran (New York University Stern School of Business) in his latest blog post. He re-examines relevant data and discusses the long-standing arguments for adding a size premium to the discount rate for small firms.
Triple whammy: He cites three reasons why he doesn't use a size premium: (1) the historical data are yielding more ambiguous results, which call into serious question whether a size premium exists; (2) forward-looking risk premiums are yielding no premiums for small-cap stocks; and (3) if based on intuition (that smaller firms are riskier than larger firms), much of the extra risk can either be diversified away, better treated as an adjustment to expected cash flows, or it's double counted.
He also makes the point that "inertia" is another culprit. Some analysts add a size premium by rote because that's the way it's always been done. And he points out that the custom is reinforced by the legal system that looks to precedence as opposed to what may be a better practice. Damodaran made this same point in an interview in Business Valuation Update (September 2014) when he said: "For better or worse, you can have bad valuations that are legally defensible and good ones that are not."
Comments: "I would highly recommend that all appraisers read Professor Damodaran's take on the small stock premium," Bob Dohmeyer (Dohmeyer Valuation Corp.) tells BVWire. "It contains very insightful commentary along with a beautifully nuanced analysis and coherent conclusion based on all of the available empirical data. Whether or not you personally agree with his findings, all appraisers should read this in order to be prepared for possible critiques. The thought of being open-minded is also important and we should keep that in mind."
"I think Professor Damodaran asks some bluntly honest questions that few have the courage or knowledge to address," says Michael A. Crain (The Financial Valuation Group). "Perhaps BV has reached a time to seriously assess some of its practices as Wall Street and regulators did on standard finance practice after the financial crisis of 2008 and 2009. This kind of discussion reminds me of a Harvard Business Review article that I read about 10 years ago by Emanuel Derman, a Ph.D. and former quant on Wall Street. He writes that modern economists tend to explain things with math even though, on average, the world is too complex for that. Contemporary finance (financial economics) research and practice tends to be part of this math phenomena."
What do you think? Send us your comments.
Delaware Chancery grapples with sale price versus post-merger DCF
What's more plausible: the merger price or the value derived from the discounted cash flow? This was the overarching question the Delaware Court of Chancery recently explored in a statutory appraisal action.
Backstory. Ancestry.com was a self-described "pioneer and the leader in the online family research market." The company went public in 2009, when its shares traded at $13.50 each. After it became the sponsor of a popular NBC show, "Who Do You Think You Are?" the price per share rose as high as $40. But its fortunes changed again, and its board decided to sell the company to a private equity investor for $32 per share. The merger price represented a 41% premium on the unaffected trading price of company stock. Ninety-nine percent of the voting shares approved of the transaction, but the dissenters asked the court for a fair value determination.
Both sides retained experts who agreed on the use of the DCF analysis but disagreed over key inputs. According to the Chancery, the petitioners' expert "proved something of a moving target" in that he proposed at different points in the litigation that the company was worth as much as $47 per share and no less than $42.81. The company's expert concluded the stock was only worth $30.63 per share. That price was actually below the value the buyer, a nonstrategic investor, was willing to pay, the court observed.
'Riffs on market price': Both experts were respected and well qualified, but their valuations were "less than fully persuasive," the Chancery noted. The analyses were "result-oriented riffs on the market price." By his own account, the petitioners' expert "tortured the numbers until they confess[ed]." The company's expert "candidly suggested" that if his valuation had been as far from the merger price as that of the petitioners' expert, he "would have tried to find out a way to reconcile those two numbers."
The Chancery said these statements showed the limited use of a post hoc DCF valuation. "If an analysis, relied upon to assess whether a sales price presents the fair value, in turn uses that very sales price as a check on its own plausibility, and if it must be revised if it fails that check, then the process itself approaches tautology."
Instead of relying on either expert's analysis completely, the court performed its own DCF, arriving at a value of $31.79 per share. But, since the sales process was "reasonable, wide-ranging and produced a motivated buyer," the court decided it would be "hubristic" to elevate its DCF estimate of value over the value an entity "for which investment represents a real—not merely an academic—risk" placed on the company. Therefore, the Chancery found the $32 merger price best represented the fair value of company stock. The DCF value was useful as a check on the market-derived valuation, the court said.
Find a discussion of In re Ancestry, 2015 Del. Ch. LEXIS 21 (Jan. 30, 2015) in the May edition of Business Valuation Update; the court's opinion is available at BVLaw.
GLOBAL BV NEWS:
Tweedie targets financial instrument valuation
In a video interview, Sir David Tweedie, chairman of the International Valuation Standards Council (IVSC), talks about the work of the IVSC and its efforts to develop unified global standards for valuation. One particular area of concern is the valuation of financial instruments.
"We are finding that financial institutions are getting quite different, wide variations in the value of these identical instruments," he says. "That instantly worries you because if some of them are hugely different, and we know they are, you have a situation where you don't know how reliable the balance sheet is. And the balance sheet clearly affects the income statement, so how reliable is the profit number? And then if we talk about financial stability, which is based on the capital buffers, based on balance sheet numbers, are those rubbish, or do they mean something?"
Tweedie is pulling together regulators, auditors, banks, and standard-setters to work toward unifying standards for financial instruments—which can be extremely complex mechanisms. "Here's the guys with the pointy heads and the whirling eyes, astrophysicists and so on," Tweedie remarks. "We have to say: 'Wait a minute. What are you doing in your little black box?' We have to bring this out and shine a light on it and see what is happening."
Extra: To reflect the growing importance of business valuation in an international setting, BVR has launched its new Global Business Valuation Resource Centre. It's designed to deliver best-in-class news, training, research, and data tools to business valuation professionals around the world. If you have ideas on topics, products, and/or news items, we would welcome the chance to speak with you! Contact Lexie Gross at lexieg@bvresources.com/+1 617 861 770.
Free download of select 2015 Mergerstat data now available
Did the total value of announced M&A deals in 2014 exceed those in 2013? What were the top five seller industries in 2014, in number and dollar value of announcements? Which U.S. states announced the greatest number of corporate sales/divestitures in 2012?
The answers to these questions—and more—are all in this new free download of selected data from the newly released 2015 Mergerstat Review, which gives you quick access to selling price multiples by industry and premiums paid by industry. Not only does the book provide a review of the 2014 M&A activity in a statistical analysis section, but it also includes:
Industry analysis for 2014;
Geographical analysis for 2014;
Details from completed and pending 2014 transactions in a transaction roster; and
Historical review going back 25 years.
You also receive a free monthly bonus: the Factset Mergerstat Monthly Review, in PDF form, that will make sure you're always up-to-date with the latest M&A news and trends, industry activity and value comparisons, private equity updates, top advisor activity, and top U.S. deals.
The 2015 Mergerstat Review is available now in PDF form. The hardcover version will be available to ship in late April. To order the PDF or preorder the hardcover version, click here.
Healthcare services multiples and trading performance
The S&P Healthcare Services Index has increased by 16% over the past three months, thus outperforming the S&P 500 (a 1.8% increase over the same period), according to the March 2015 Healthcare Sector Update from Duff & Phelps. The best performing sectors were emergency services (up 32.5%) and managed care—government (up 24.5%). The worst performing sectors were specialty managed care (down 13.9%) and physician practice management (down 0.5%).
Latest multiples: The current median LTM revenue and LTM EBITDA multiples for the healthcare services industry overall are 1.75x and 12.1x, respectively. The sectors with the highest valuation multiples include: healthcare REITs (12.96x LTM revenue, 20.1x LTM EBITDA) and HCIT (3.49x LTM revenue, 21.39x LTM EBITDA).
The report also provides data on the pharmaceutical/medical devices/life sciences sectors.
Trademark valuation data for CRM software
The average trademark royalty rate for CRM software is 2% of revenue, according to a new analysis from Markables. This information is gleaned from the Markables database of over 5,000 trademark valuations published in financial reporting documents of listed companies from all over the world. The database reports value solely for the use of trademarks (not bundled with other rights).
Preview of the May issue of Business Valuation Update
Here's what you'll see:
Indefinite Is Not Infinite—Solving a Dichotomy in Trademark Valuation (Christof Binder, Ph.D., MBA, and Robert B. Morrison, MBA, ASA BVIA). New research reveals that trademark valuations appear to be not fully in line with ASC 350 and IAS 38. Once a trademark is given an indefinite life, it appears to be treated that way permanently, which has serious effects on valuation and accounting. The authors also discuss how to estimate a finite remaining useful life of a trademark.
Misapplication of the Gordon Growth Model Can Lead to Undervaluing (Drew E. Voth, CVA, CPA/CFF, CIRA/CDBV, CFE). Every valuation analyst is taught the Gordon growth model, and it's not particularly complex. But the author sees common mistakes being made in applying this model and looks behind the model to demonstrate its proper application.
Avoiding Complications and Disparate Opinions of Value in Buy-Sell Agreements (Brian Burns, CPA/ABV/CFF, ASA, and Chris Mitchell, CPA/ABV/CFF). Common ambiguities and valuation pitfalls can lead to complications surrounding buy-sell agreements. Also, relevant case law concerning disputes with buy-sell agreements is discussed.
BVU Profile: Business Valuation From the Perspective of an Appraiser in Italy (BVR Editor). An interview with Ascanio Salvidio (Salvidio & Partners), who talks about the valuation profession in Italy and the importance of a convergence of valuation standards. He also discusses the data sources he uses for valuation purposes.
Ongoing Research Related to the S Corp Valuation Puzzle (BVR Editor). Nancy Fannon (Meyers, Harrison and Pia LLC) and Keith Sellers (University of Denver) present several areas where additional research would be helpful to further the analysis concerning the effect of shareholder-level taxes on a firm's value. Their new book is a major advance in the area of valuations of pass-through entities.
To read these articles—as well as digests of the latest court cases—see the May issue of Business Valuation Update (subscription required).
BV movers . . .
People: John Bautista joins CohnReznick as a principal in the firm's Valuation Advisory Services Practice and will be based out of the New York City office … Brian Kerwin was named chair of corporate practice group of Duane Morris and will continue to work out of the Chicago office, which he helped establish in 1999 … Mark Meinberg was named partner-in-charge of EisnerAmper's Long Island office in Syosset, N.Y., and succeeds Eric Altstadter, who for the past five years has led this office's growth from half a dozen staffers to nearly 40 … The Central Penn Business Journal named Bethany Novis, a managing partner at Reinsel Kuntz Lesher, a "2015 Woman of Influence."
Firms: Pittsburgh-based The Meridan Group, an investment banking and management consulting firm specializing in business advisory services and M&A consulting, has merged with the national firm Schneider Downs … Porte Brown was chosen as one of the "2015 Best Places to Work" in Illinois … Stockman Kast Ryan + Co was named the exclusive Colorado Springs, Colo., member of the nationwide association of Auto Dealer CPAs, where membership is based on a demonstrated proficiency and commitment to the auto dealer industry.
CPE events
Put the tax season behind you and check out this exciting slate of CPE events:
Valuation's Orphan: Economic Benefit Streams (April 21), with Darrell Dorrell (Financial Forensics);
To Err is Human: Avoiding Common Valuation Mistakes (April 23), with Kevin Yeanoplos (Brueggeman and Johnson Yeanoplos, P.C.);
Tax Issues Arising in Valuations (April 28), with Heather Tullar and Patrice Riela (both with Delphi Valuation Advisors Inc.); and
Law Firms: Valuation and Other Consulting Opportunities (April 29), with Ronald Seigneur (Seigneur Gustafson LLP).
Important note to webinar attendees: To ensure that you receive your dial-in instructions to BVR's training events, please make sure to whitelist bvreducation@bvresources.com.
We welcome your feedback and comments. Contact the editor, Andy Dzamba at:
info@bvresources.com or (503) 291-7963 ext. 133
Not a BVWire subscriber?
Get on the list today!
Size premium under fire
Post-merger DCF
Global BV News
2014 M&A data
Healthcare multiples
Trademark values
May BVU preview
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redpajama_set_name: RedPajamaCommonCrawl


The Frankfurt Zoological Society (FZS) is an international conservation or- ganisation based in Germany. FZS is active in biodiversity-rich areas with projects on four continents. The Africa Programme is the organisation's largest and oldest and includes projects in five focal countries: Ethiopia, DR Congo, Tanzania, Zambia and Zimbabwe. In the Democratic Republic of the Congo, FZS has previously supported the national protected area agency, L'Institut Congolais pour la Conservation de la Nature (ICCN) in Virunga, Maiko and Upemba National Parks.
The TL2 Project is an initiative of the Lukuru Foundation that supports conservation in and around Lomami National Park (PNL) in DR Congo.
As the Programme Leaders anticipate their retirement in the next few years, FZS is assuming responsibility for support to PNL in the long-term and gradually taking over all project activities from the Lukuru Foundation/ TL2. To ensure a smooth transition from the Lukuru Foundation to FZS and strong leadership for the existing team in PNL after the transition, FZS is looking for a highly motivated Project Leader to work in and around PNL.
• Manage all aspects of the project in PNL, including staff, budgets, activities and planning. Initially this will be in close collaboration with, and under the guidance of, the current TL2 Programme Leaders.
• Conduct work planning, budgeting, performance appraisals, staff coaching and other tasks associated with project leadership, as per FZS guidance.
• Assume responsibility in a co-management role in partnership with ICCN as and when this becomes relevant. This will involve more direct management and decision-making related to PNL.
An attractive salary package based on our in-country or international em- ployment schemes. An initial two-year contract is standard practice, with the perspective of a long-term position in an internationally recognized conservation organization. Probation period is 6 months.
Please send your application including a cover letter including your salary expectations and your earliest possible starting date and a chronological CV in PDF format to jobs@fzs.org.
1. What led you to apply for this position?
2. What is the most important technical ability you would bring to the role?
3. What is the most important technical ability that you would need to develop to successfully fulfil this role?
4. What is your most important personal characteristic that would help you succeed in this role?


text:
Q: HashCodeBuilder and EqualsBuilder usage style I often use apache HashCodeBuilder and EqualsBuilder for object equality using reflection, but recently I a colleague told me that using reflection may cause a huge performance hit if the entity contains lots of properties. Worried that I may be using a wrong implementation, my question is, which of the following approach would you prefer? And why?
public class Admin {

    private Long id;
    private String userName;

    public String getUserName() {
        return userName;
    }

    @Override
    public boolean equals(Object o) {
        if (!(o instanceof Admin)) {
            return false;
        }
        Admin otherAdmin  = (Admin) o;
        EqualsBuilder builder = new EqualsBuilder();
        builder.append(getUserName(), otherAdmin.getUserName());
        return builder.isEquals();
    }

    @Override
    public int hashCode() {
        HashCodeBuilder builder = new HashCodeBuilder();
        builder.append(getUserName());
        return builder.hashCode();
    }
}

Vs.
public class Admin {

    private Long id;
    private String userName;

    public String getUserName() {
        return userName;
    }

    @Override
    public boolean equals(Object o) {
      return EqualsBuilder.reflectionEquals(this, o, Arrays.asList(id));
    }

    @Override
    public int hashCode() {
        return HashCodeBuilder.reflectionHashCode(this, Arrays.asList(id));
    }
}

A: Your question as written clearly illustrates one of the benefits of the second approach:
In the first case, it's very easy to make the mistake return builder.hashCode(), instead of the correct return builder.toHashCode(), which will result in subtle errors that can be very hard to track down.
The second case eliminates the possibility of this typo, resulting in less banging your head into the keyboard trying to find the bug.

A: I would say neither one of these are good implementation. I would argue that EqualsBuilder is not a good framework to use for the following reasons:

*

*Not extendable. What if one of the field you are trying to assert equality should treat nulls and blank as equal?

*You must maintain the list of variable as if they are hardcoded variables. Meaning you must list all the variables you want to compare. At this point there isn't any different between a == o.getA() && b == o.getB() ...

*Using reflection takes up additional resource as you have pointed out, and in an enterprise application that crush billions of objects. Doing this reflection equals is as bad as having a memory leak.

I'll say there needs to be a better framework than the Apache one.

A: Of course the second option is more elegant and simple. But if you are concerned about performance you should go for first approach. Second method also fails if a security manager is running.
I would go for the first option if I was in your situation.
Also there is a mistake in your first approach in generating hashCode:
It should be builder.toHashCode()
instead of builder.hashCode(). The latter returns hashcode builder object's hash code.

A: Agree with @Churk, Apache HashCodeBuilder and EqualsBuilder are not implemented well. HashCodeBuilder is still playing with prime numbers!  Further, it does an awful lot of unnecessary work.  Have you read the source?
Since Java 5 (if not earlier), AbstractHashMap<> has not used modulo of a prime number to locate the hash bucket.   Instead, the number of buckets is a power of two and the low order N bits of the hash code is used to locate the bucket.
Further, it will "mix" the hash code supplied by the application so that the bits are spread evenly and, thus, the buckets are filled evenly.
Thus, the correct way to override int Object.hashCode() is by returning the simplest, constant value with the highest arity in the population of objects that will cohabitate in any collection using the class.
Typically, the ID value unmodified is your best bet.   If your ID field is integral, just cast it to (int) and return it.   If it is a String or other Object, just return its hash code.  You get the idea.  For a compound identifier, return the field (or hashCode thereof) with the most distinct values.  Less is more.
Of course, the contract between hashCode() and equals() must be met.  Thus, equals() should be implemented accordingly.   hashCode() need not use the full qualifiers needed for equality, but any fields used in hashCode() must be used in equals().   Here, methods like StringUtils.equals( s1, s2 ) are useful for handling the null values consistently and safely.

A: Even though the second option is more attractive (because it is just one line of code) I would choose the first option.
The reason is simply performance. After running a small test I found a very great time difference between them.
In order to sort of get an idea of time, I created this two simple classes:
package equalsbuildertest;

import java.math.BigDecimal;
import java.util.Date;

import org.apache.commons.lang3.builder.EqualsBuilder;
import org.apache.commons.lang3.builder.HashCodeBuilder;

public class Class1 {

    private int field1;

    private boolean field2;

    private BigDecimal field3;

    private String field4;

    private Date field5;

    private long field6;

    public Class1(int field1, boolean field2, BigDecimal field3, String field4,
            Date field5, long field6) {
        super();
        this.field1 = field1;
        this.field2 = field2;
        this.field3 = field3;
        this.field4 = field4;
        this.field5 = field5;
        this.field6 = field6;
    }

    public Class1() {
        super();
    }

    public int getField1() {
        return field1;
    }

    public void setField1(int field1) {
        this.field1 = field1;
    }

    public boolean isField2() {
        return field2;
    }

    public void setField2(boolean field2) {
        this.field2 = field2;
    }

    public BigDecimal getField3() {
        return field3;
    }

    public void setField3(BigDecimal field3) {
        this.field3 = field3;
    }

    public String getField4() {
        return field4;
    }

    public void setField4(String field4) {
        this.field4 = field4;
    }

    public Date getField5() {
        return field5;
    }

    public void setField5(Date field5) {
        this.field5 = field5;
    }

    public long getField6() {
        return field6;
    }

    public void setField6(long field6) {
        this.field6 = field6;
    }

    @Override
    public boolean equals(Object o) {
      return EqualsBuilder.reflectionEquals(this, o);
    }

    @Override
    public int hashCode() {
        return HashCodeBuilder.reflectionHashCode(this);
    }

}

And:
package equalsbuildertest;

import java.math.BigDecimal;
import java.util.Date;

import org.apache.commons.lang3.builder.EqualsBuilder;
import org.apache.commons.lang3.builder.HashCodeBuilder;

public class Class2 {

    private int field1;

    private boolean field2;

    private BigDecimal field3;

    private String field4;

    private Date field5;

    private long field6;

    public Class2(int field1, boolean field2, BigDecimal field3, String field4,
            Date field5, long field6) {
        super();
        this.field1 = field1;
        this.field2 = field2;
        this.field3 = field3;
        this.field4 = field4;
        this.field5 = field5;
        this.field6 = field6;
    }

    public Class2() {
        super();
    }

    public int getField1() {
        return field1;
    }

    public void setField1(int field1) {
        this.field1 = field1;
    }

    public boolean isField2() {
        return field2;
    }

    public void setField2(boolean field2) {
        this.field2 = field2;
    }

    public BigDecimal getField3() {
        return field3;
    }

    public void setField3(BigDecimal field3) {
        this.field3 = field3;
    }

    public String getField4() {
        return field4;
    }

    public void setField4(String field4) {
        this.field4 = field4;
    }

    public Date getField5() {
        return field5;
    }

    public void setField5(Date field5) {
        this.field5 = field5;
    }

    public long getField6() {
        return field6;
    }

    public void setField6(long field6) {
        this.field6 = field6;
    }

    @Override
    public boolean equals(Object obj) {
        if (!(obj instanceof Class2)) {
            return false;
        }
        Class2 other = (Class2) obj;
        EqualsBuilder builder = new EqualsBuilder();
        builder.append(field1, other.field1);
        builder.append(field2, other.field2);
        builder.append(field3, other.field3);
        builder.append(field4, other.field4);
        builder.append(field5, other.field5);
        builder.append(field6, other.field6);
        return builder.isEquals();
    }

    @Override
    public int hashCode() {
        HashCodeBuilder builder = new HashCodeBuilder();
        builder.append(getField1());
        builder.append(isField2());
        builder.append(getField3());
        builder.append(getField4());
        builder.append(getField5());
        builder.append(getField6());
        return builder.hashCode();

    };

}

That second class is pretty much the same as the first one, but with different equals and hashCode.
After that, I created the following tests:
package equalsbuildertest;

import static org.junit.Assert.*;

import java.math.BigDecimal;
import java.util.Date;

import org.junit.Test;

public class EqualsBuilderTest {

    @Test
    public void test1() {
        Class1 class1a = new Class1(1, true, new BigDecimal(0), "String", new Date(), 1L);
        Class1 class1b = new Class1(1, true, new BigDecimal(0), "String", new Date(), 1L);
        for (int i = 0; i < 1000000; i++) {
            assertEquals(class1a, class1b);
        }
    }

    @Test
    public void test2() {
        Class2 class2a = new Class2(1, true, new BigDecimal(0), "String", new Date(), 1L);
        Class2 class2b = new Class2(1, true, new BigDecimal(0), "String", new Date(), 1L);
        for (int i = 0; i < 1000000; i++) {
            assertEquals(class2a, class2b);
        }
    }

}

The test are pretty simple and only serves to measure time.
The results were the following:

*

*test1 (2,024 s)

*test2 (0,039 s)
I chose them to be completely equal in order to have the greatest times. If you choose to do the test with NotEquals conditions you will have shorter time, but keeping a very large time difference too.
I run this tests on a 64-bit Intel Core i5-3317U CPU @1.70GHz x4 with Fedora 21 and Eclipse Luna.
In conclusion, I would not risk such a great performance difference in order to save a couple of lines of code that you can possibly not type anyway using a template (in Eclipse under Windows -> Preferences is found in Java -> Editor -> Templates) such as this:
${:import(org.apache.commons.lang3.builder.HashCodeBuilder, org.apache.commons.lang3.builder.EqualsBuilder)}
@Override
public int hashCode() {
    HashCodeBuilder hashCodeBuilder = new HashCodeBuilder();
    hashCodeBuilder.append(${field1:field});
    hashCodeBuilder.append(${field2:field});
    hashCodeBuilder.append(${field3:field});
    hashCodeBuilder.append(${field4:field});
    hashCodeBuilder.append(${field5:field});
    return hashCodeBuilder.toHashCode();
}

@Override
public boolean equals(Object obj) {
    if (this == obj) {
        return true;
    }
    if (obj == null) {
        return false;
    }
    if (getClass() != obj.getClass()) {
        return false;
    }
    ${enclosing_type} rhs = (${enclosing_type}) obj;
    EqualsBuilder equalsBuilder = new EqualsBuilder();
    equalsBuilder.append(${field1}, rhs.${field1});
    equalsBuilder.append(${field2}, rhs.${field2});
    equalsBuilder.append(${field3}, rhs.${field3});
    equalsBuilder.append(${field4}, rhs.${field4});
    equalsBuilder.append(${field5}, rhs.${field5});${cursor}
    return equalsBuilder.isEquals();
}

A: I would prefer the second option for 2 reasons:

*

*Obviously it is easier to read

*I wouldn't buy performance arguments for the first option, unless those include a relevant metric.
E.g. how many milliseconds would reflection-based "equals" add to a typical end-to-end request latency? Overall, what % increase would that be?
Without knowing that good chances are that the optimization is premature

redpajama_set_name: RedPajamaStackExchange


text:
\section{Introduction}

Recently, the possibility to  induce an out-of-equilibrium topological state of matter through irradiation or a periodic driving has stimulated numerous works. While initially
 the external driving perturbation was used to trigger a  phase transition between states of conventional topological order \cite{Inoue:2010,Lindner:2011,Kitagawa:2011},
 fascinating topological properties specific to driven out-of-equilibrium states were soon identified \cite{Kitagawa:2010,Rudner:2013,Carpentier:2015}.
While several proposals to realize and probe these topological states in various artificial systems have turned out to be successful
\cite{Fang:2012,Kitagawa:2012,Hauke:2012,Rechtsman:2013,Hu:2015,Karzig:2014,Reichl:2014,Jotzu:2014}.
Their realization  in condensed matter have proved to be challenging \cite{Wang:2013,Onishi:2014}.

There is a strong analogy between equilibrium topological insulators and topological driven states. Both require the existence of a gap in
the spectrum characterizing their single particle states: topological insulators are band insulators with a gap in the energy spectrum of the single particle
Hamiltonian while topological driven states have a gap in the spectrum of the Floquet operator.
In both cases,  a nontrivial topology manifests itself through the appearance within this gap of robust states located at the edge of the system.
However, while in an insulator the gap separates empty states from occupied states,
the thermodynamics of gapped periodically driven states is much less understood. Recent studies
have stressed the differences between the nature of the states reached at long time in such periodically driven systems and the equilibrium
ground states of insulators \cite{Lazarides:2014,Dehghani:2014,Lacedola:2015,Seetharam:2015}.

Here we follow a different route: we focus on the relation between the DC transport of a periodically driven system and its quasi-energy Floquet
spectrum in a regime where the times of flight of electrons through the system are shorter than the characteristic inelastic scattering times, which can be the case in mesoscopic systems. This provides a way to avoid the issue of long time dynamics of driven systems, which was raised in recent studies~\cite{Lacedola:2015,Seetharam:2015,Dehghani:2015}.
Technically, this requires that the dominant perturbation of the unitary evolution of the driven system is the presence of the electrodes: \emph{dissipation
should occur in the leads}.
From this point of view, the driven system behaves as a topological Floquet filter instead of an out-of-equilibrium steady-state analogue of an equilibrium insulating state.

DC transport is known to be an ideal  probe of the existence of edge states in topological equilibrium phases
realized in condensed matter, particularly in two dimensions.
In a seminal paper \cite{Buttiker:1988} Markus Büttiker demonstrated how the non-local conductances in a Hall bar fully characterize
the nature of the quantum Hall effect and the associated chiral edge states.
This approach was recently extended to study the quantum spin Hall effect occuring in HgTe/CdTe quantum wells \cite{Roth:2009,buttiker2009}. In this
time-reversal invariant topological phase, the existence of a Kramers pair of counter-propagating edge states leads to a series of non-local conductances
whose experimental observation clearly identified this new phase.
For topological driven systems, the situation is more confusing:
building on earlier works on the transport through a topological periodically driven state  \cite{Gu:2011,Kitagawa:2011}, recent studies have focused on
the transport through a one-dimensional topological superconducting state \cite{Kundu:2013},
the effect on transport of the competition between heating by the drive and the coupling to the leads  \cite{Dehghani:2015} or
the quantization of conductances of a topological phase in multi-terminal geometry~\cite{FoaTorres:2014}.
It was also proposed to probe quasienergy spectra (and topological edge states) through magnetization measurements \cite{dahlhaus2015} and tunneling spectroscopy \cite{fregoso2014}.
However, the relation between
transport and the existence of topological edge states in periodically driven states remains unclear, and a summation procedure over different energies in
the lead was proposed to recover a quantized conductance  \cite{Kundu:2013,Farrell:2015}.
The purpose of our paper is to reconsider the relation between the (non-local) differential conductances of periodically driven systems and
their Floquet quasi-energy spectrum, allowing for a direct relation between these differential conductances and the topological indices associated with the
spectral gaps. In particular we will establish a protocol in a multi-terminal geometry allowing for this identification.
In this point of view, a topological periodically driven system is viewed as a
\emph{topological Floquet filter} with selective edge transport occurring for specific voltage biases between a lead and the system. These voltage biases lead to a stationnary DC current by counterbalancing the time dependence of Floquet states.

\section{From Floquet theory to scattering theory}

\subsection{Floquet theory for open systems}

We consider a periodically driven quantum system connected to $N_{\text{leads}}$ equilibrium electrodes through good contacts with large transmissions.
The system is described by a Hamiltonian $\hat{H}^{\text{sys}}(t) - \hat{\Sigma} $ where
$\hat{H}^{\text{sys}}(t+T)=\hat{H}^{\text{sys}}(t)$ with $T$  the period of the drive, and $ \hat{\Sigma}$ is a self-energy accounting for the
coupling between the system and its environment (e.g. the leads). We assume in the following that this self-energy is dominated by the exchange with the
electrons in the leads. When all characteristic times of the leads are small with respect to the characteristic times of the system, we can use the so-called wide band approximation \cite{Kohler:2005} where the self-energy is assumed to be constant in energy: $\hat{\Sigma}(E) \simeq \hat{\Sigma}$.
The dynamics of the system is described by the evolution operator $\hat{U}(t,t')$ which obeys the equation
\begin{equation}
	\ensuremath{\mathrm{i}} \hbar \frac{\ensuremath{\mathrm{d}}}{\ensuremath{\mathrm{d}} t}\hat{U}(t,t') = \left( \hat{H}^{\text{sys}}(t) -  \hat{\Sigma} \right) \hat{U}(t,t') .
\label{eq:schrodinger_U}
\end{equation}
Of great importance is the \textit{Floquet operator} which is the evolution operator after one period $\hat{U}(T,0)$.
When diagonalizable, it can be decomposed on the left eigenstates  $\bra{\tilde{\phi}_\alpha}$  and the right eigenstates  $\ket{\tilde{\phi}_\alpha}$ of $\hat{U}(T,0)$
\begin{align}
 \begin{split}
	\hat{U}(T,0) \ket{\phi_\alpha} &= \lambda_\alpha \ket{\phi_\alpha} , \\
	\bra{\tilde{\phi}_\alpha}\hat{U}(T,0) &= \lambda_\alpha \bra{\tilde{\phi}_\alpha} ,
\end{split}
\label{eq:lambda}
\end{align}
that constitute a bi-orthonormal basis of the Hilbert space
\begin{equation}
	\braket{\tilde{\phi}_\alpha \mid \phi_\beta} = \delta_{\alpha \beta} \ ; \
	\sum_{\alpha} \ket{\phi_\alpha}\bra{\tilde{\phi}_\alpha} = \ensuremath{\text{Id}} .
\label{eq:basis}
\end{equation}
The eigenvalues $\lambda_\alpha$ in Eq.~\eqref{eq:lambda} are called the Floquet multiplicators and read
\begin{equation}
	\lambda_\alpha = \exp\left[ - \ensuremath{\mathrm{i}} \left( \frac{\varepsilon_\alpha}{\hbar}  - \ensuremath{\mathrm{i}} \gamma_\alpha \right) T \right].
\end{equation}
The coefficient $ \varepsilon_{\alpha}$ is called the quasienergy and $\gamma_{\alpha}$ is its damping rate whose inverse gives the life-time of the eigenstate.
Note that the quasienergy being a phase, it is defined modulo the driving frequency $\omega=2\pi/T$.
Any state at arbitrary time $t$ can then be constructed from the eigenstates of the Floquet operator.
It is particularly useful to define the left and right Floquet states
\begin{align}
 \begin{split}
	\ket{u_\alpha(t)} &= \ensuremath{\mathrm{e}}^{ \ensuremath{\mathrm{i}} \left( \varepsilon_\alpha/\hbar  - \ensuremath{\mathrm{i}} \gamma_\alpha \right) t} \   \hat{U}(t,0)\ket{\phi_\alpha}  , \\
	\bra{\tilde{u}_\alpha(t)} &= \ensuremath{\mathrm{e}}^{- \ensuremath{\mathrm{i}} \left( \varepsilon_\alpha/\hbar  - \ensuremath{\mathrm{i}} \gamma_\alpha \right) t}\ \bra{\tilde{\phi}_\alpha}  \hat{U}(0,t) ,
\end{split}
\label{eq:u_def}
\end{align}
which are periodic in time, $\ket{u_\alpha(t)}=\ket{u_\alpha(t+T)}$ (same for $\bra{\tilde{u}_\alpha(t)}$), so that they can be expanded in Fourier series
\begin{align}
 \begin{split}
	\ket{u_{\alpha}(t)} &= \sum_{p \in \mathbb{Z}} \ensuremath{\mathrm{e}}^{-\ensuremath{\mathrm{i}} p \omega t} \ket{u_{\alpha}^{(p)}} , \\
	\bra{\tilde{u}_{\alpha}(t)} &= \sum_{p \in \mathbb{Z}} \ensuremath{\mathrm{e}}^{\ensuremath{\mathrm{i}} p \omega t} \bra{\tilde{u}_{\alpha}^{(p)}} ,
\label{eq:u_fourier}
\end{split}
\end{align}
where the harmonics read
\begin{align}
 \begin{split}
 \ket{u_{\alpha}^{(p)}} = \frac{1}{T} \int_{0}^{T} \ensuremath{\mathrm{d}} t \; \ensuremath{\mathrm{e}}^{\ensuremath{\mathrm{i}} p \omega t} \ket{u_{\alpha}(t)} \\
 \bra{\tilde{u}_{\alpha}^{(p)}} = \frac{1}{T} \int_{0}^{T} \ensuremath{\mathrm{d}} t \; \ensuremath{\mathrm{e}}^{-\ensuremath{\mathrm{i}} p \omega t} \bra{\tilde{u}_{\alpha}(t)} .
\end{split}
\label{eq:harmonics}
\end{align}
From  Eqs. (\ref{eq:basis}) and (\ref{eq:u_def})  the evolution operator can be expanded on the Floquet states as
\begin{equation}
\hat{U}(t, t') = \sum_{\alpha} \ensuremath{\mathrm{e}}^{-\ensuremath{\mathrm{i}} \left( \varepsilon_\alpha/\hbar  - \ensuremath{\mathrm{i}} \gamma_\alpha  \right)\left(t-t' \right)} \ket{u_{\alpha}(t)}\bra{\tilde{u}_{\alpha}(t')} .
\end{equation}
This expression can finally be decomposed on the harmonics of the Floquet states by using Eq.(\ref{eq:u_fourier})
\begin{multline}
	\label{eq:U_decomposition}
	\!\!\!\!\hat{U}(t, t') = \!\!\!
	\sum_{\substack{\alpha \\ p,p' \in \mathbb{Z}}} \!\!\! \ensuremath{\mathrm{e}}^{-\ensuremath{\mathrm{i}} \left[ \left( \frac{\varepsilon_\alpha}{\hbar}  - \ensuremath{\mathrm{i}} \gamma_\alpha  \right)(t-t') + \omega (p t - p' t')\right]}
	\ket{u_{\alpha}^{(p)}}\!\bra{\tilde{u}_{\alpha}^{(p')}}.
\end{multline}

In practice, the spectrum of Floquet operator of the semi-infinite system can be obtained numerically either by direct a computation of $U(T,0)$ (e.g. as a discretized in time version of the infinite product) or through its representation in Sambe space \cite{sambe73}.

\subsection{Differential conductance}

Based on a standard formalism, we can calculate analytically the differential conductance of the periodically driven system in a multi-terminal geometry
and relate it to the quasienergy spectrum of the system.
We follow the standard Floquet scattering formalism  \cite{Moskalets:2002,Kohler:2005,Stefanucci:2008,Gaury:2013,Kundu:2013,FoaTorres:2014} to describe the
transport properties of this multiterminal setup in a phase coherent regime.
We consider the rolling average over a period~$T$ (all time-averages in the following are also rolling averages over one driving perdiod) of the current entering each lead labelled by the index~$\ell$:
\begin{equation}
	\mathcal{I}_{\ell}(t) =  \frac{1}{T} \int_{t}^{t+T} \ensuremath{\mathrm{d}} t' \; \braket{\hat{J}_{\ell}(t')}.
	\label{eq:averagedCurrent}
\end{equation}
where $\braket{\hat{J}_{\ell}(t')}$ is the expectation value of the current entering lead~$\ell$ at time~$t'$.
This average current satisfies a relation~\cite{Moskalets:2002,Kohler:2005,Stefanucci:2008}:
\begin{equation}
	\mathcal{I}_{\ell}(t) = \frac{e}{h}  \int \ensuremath{\mathrm{d}}E\;
 	\sum_{\ell' \neq \ell} \left[
	 T_{\ell \ell'}(t, E) f_{\ell'}(E)
	  - T_{\ell' \ell}(t, E) f_{\ell}(E)
	\right],
	\label{eq:Il_stat}
\end{equation}
where $f_{\ell}(E)$ is the Fermi-Dirac distribution of the lead $\ell$ assumed to be at equilibrium at the chemical potential $\mu_{\ell}$.
The $T_{\ell \ell'}(E) $ are the time-averaged transmission coefficients between lead $\ell$ and $\ell'$ which will be discussed below.
We define the differential conductance $G_{\ell \ell'}(t, E)$ as the sensitivity of the current entering the lead $\ell$ to variations of the chemical potential $\mu_{\ell'}$
of the lead\footnote{See \cite{Gaury:2013} for a recent discussion  of chemical versus electrical potential drops at the interface between the system and an electrode. Such
subtleties are quietly neglected in the following.} $\ell'$ :
\begin{equation}
        G_{\ell \ell'} (t, E)\equiv - e \left. \frac{\ensuremath{\mathrm{d}} \mathcal{I}_{\ell}}{\ensuremath{\mathrm{d}} \mu_{\ell'}}  \right|_{\mu_{\ell'}=\mu_{\text{sys}}+E}.
	\label{eq:diff_cond}
\end{equation}
Note that this definition is not symmetric in the various chemical potentials $\mu_{\ell}$, as opposed to other definitions used in the literature.
In the long time stationary regime on which we focus, $t \to \infty$, the average conductances $G_{\ell \ell'}(t,E)$ are expected to reach a value independent on the choice of origin of time~$t$ and associated initial conditions \cite{Kohler:2005}, which we denote by $G_{\ell \ell'}^{\infty}(E)$.

We obtain from Eq.~\eqref{eq:Il_stat} the zero temperature time-averaged differential conductances
\begin{align}
        G_{\ell \ell'}^{\infty}(E) &= \frac{e^2}{h}  T_{\ell \ell' }(E) \qquad \text{for $\ell \neq \ell'$}
	\label{eq:diff_condllp}
\\
        G_{\ell \ell}^{\infty}(E) &= - \frac{e^2}{h} \sum_{\ell' \neq \ell}  T_{\ell' \ell }(E),
	\label{eq:diff_condll}
\end{align}
which satisfy the rule $\sum_{\ell} G_{\ell ,\ell'}(E) = 0$  for any $ \ell'$ (the average current leaving the system is independent on the chemical potentials in the leads).
This formula is analogous to the Landauer-B\"uttiker formula for
the differential conductance of multiterminal equilibrium systems.

\subsection{Generalized Fisher-Lee relations}

The average transmission coefficients $T_{\ell \ell'}(E)$ can be related to the Floquet-Green functions of the system, in a way analogous to the
case of undriven conductors~\cite{Fisher:1981}.
The retarded Green functions in mixed energy-time representations are defined by
\begin{equation}
{\hat{\mathcal{G}}}(t,E) = \frac{1}{\ensuremath{\mathrm{i}} \hbar}\int d\tau \ e^{\ensuremath{\mathrm{i}} E\tau/\hbar} \,  \hat{\cal G}(t,t-\tau)
\end{equation}
where $\hat{\cal G}(t,t')$ satisfies
\begin{align}
 \begin{split}
\label{eq:def_gf}
	\left( \ensuremath{\mathrm{i}} \hbar \frac{\ensuremath{\mathrm{d}}}{\ensuremath{\mathrm{d}} t} - \hat{H}^{\text{sys}}(t)  + \hat{\Sigma} \right) \hat{\cal G}(t,t') = \delta(t-t').
 \end{split}
\end{align}
 It is  related to the evolution operator defined in Eq.(\ref{eq:schrodinger_U}) by
\begin{equation}
 {\hat{\cal G}}(t,t')=\frac{1}{\ensuremath{\mathrm{i}} \hbar}\Theta(t-t')\hat{U}(t,t') .
\end{equation}
From the decomposition Eq.(\ref{eq:U_decomposition}) of $\hat{U}(t,t')$ over the harmonics of the Floquet states,
we  express the \textit{Floquet-Green functions}
\begin{equation}
	\label{eq_green_floquet}
	\hat{\mathcal{G}}^{(p)}(E) = \frac{1}{T} \int_{0}^{T} \ensuremath{\mathrm{d}} t\; \ensuremath{\mathrm{e}}^{\ensuremath{\mathrm{i}} p \omega t} \, {\mathcal{G}}(t, E) .
\end{equation}
as
\begin{equation}
	\label{eq_green_floquet_evolution}
	\hat{\mathcal{G}}^{(p)}(E) =
	 \sum_{r, \alpha} \frac{\ket{u_{\alpha}^{(p+r)}}\bra{\tilde{u}_{\alpha}^{(r)}} }{E - \left[ \varepsilon_\alpha + r \hbar \omega - \ensuremath{\mathrm{i}} \hbar \gamma_\alpha \right]}.
\end{equation}

The transmission coefficients are expressed as
\begin{equation}
	T_{\ell' \ell}(E)	= \sum_{p \in \mathbb{Z}} T_{\ell' \ell}^{(p)}(E) ,
	\label{eq:Tbar}
\end{equation}
where each  $T^{(p)}_{\ell' \ell }(E)$ is the transmission coefficient for an electron injected in lead $\ell$ at the energy $E=\mu_{\ell}$ and
leaving the system in lead $\ell'$ at the energy $E+p \hbar \omega$, {\it i.e.}  after having  exchanged $p$ quanta with the driving perturbation.
They read  (see \ref{sec:appendix} and \cite{Kohler:2005,Stefanucci:2008,Gaury:2013,arrachea2006}):
\begin{equation}
	\label{eq:coefficient_transmission_general}
	T_{\ell' \ell}^{(p)}(E) = \text{Tr}
	\left[ \hat{\mathcal{G}}^{{\dagger}(p)}(E) \hat{\Gamma}_{\ell'}(E + p \hbar \omega) \hat{\mathcal{G}}^{(p)}(E) \hat{\Gamma}_{\ell}(E) \right]
\end{equation}
where $\hat{\Gamma}_{\ell}(E)$ is the coupling operator at energy $E$ between the system and the electrode $\ell$.

Note that when the rates $\gamma_\alpha$ are sufficiently small for the quasi-energies $\epsilon_\alpha$ occurring in Eq.~\eqref{eq_green_floquet_evolution}
to be \enquote{well-defined}, the equations (\ref{eq_green_floquet_evolution},\ref{eq:Tbar},\ref{eq:coefficient_transmission_general}) imply that the
transmission coefficient $T_{\ell' \ell}(E))$ and thus the differential conductance $G_{\ell' \ell}(E)$ vanishes whenever the
energy $\mu_{\ell}=E$ does not correspond to a quasi-energy $\epsilon_\alpha$ up to a multiple of~$\hbar \omega$.  This is nothing but the conservation of energy
of incident state which holds modulo $\hbar \omega$ in a periodically driven system.
 This property allows one to probe the existence of gaps in the quasi-energy spectrum of the Floquet operator.
  Note however that to access the whole quasi-energy spectrum, the lead $\mu_{\ell}$ has to be strongly biased (with a bias of order~$\hbar \omega$) with respect to the
  scattering region at $\mu_{\text{sys}}$, a situation far from the standard procedure to probe equilibrium phases, but required to probe an inherently out-of-equilibrium
  Floquet state.
Moreover, in the presence of a ballistic mode connecting two leads $\ell$ and $\ell'$ such as the edge mode of a topological Floquet state,
 we expect the corresponding $T_{\ell' \ell}(E)$ to be quantized \emph{provided both interfaces are sufficiently transparent}. Note however
 that as the states leaving the systems have component of energies $E+p\hbar \omega$, this implies a perfect transparency over a broad range of energies.
 These two properties allow for a potential probe of topological Floquet states through non-local transport.
 In the next section, we study this application by a numerical implementation of transport through an AC driven system.

\section{Numerical study}

In the following, we provide numerical evidence of the correspondence between the differential conductance and the quasi-energy spectrum in a periodically driven system. In particular, we show that a single out-of-equilibrium topological edge state corresponds to a quantized differential conductance.
The chirality of these topological edge states is probed in a multi-terminal setup.

\subsection{Model and method}
\begin{figure}
	\centering
	\pgffig{reseau}
	\vspace*{-0.25cm}
	\caption{\label{fig:reseau} Lattice of couplings for the half-BHZ model with two orbitals per site of the square lattice.
	Blue and red links correspond to the $ T^{(1)}_x$ and $ T^{(1)}_y$ nearest neighbor
	coupling matrices between $s$ and $p$ orbitals while the green links represent  $T^{(2)}$ the second-nearest neighbor coupling matrix. }
\end{figure}
\begin{figure*}[t]
\centerline{
	\pgffig{fig_qe_spectrum_conductance_trivial}
	\pgffig{fig_qe_spectrum_conductance_topological}
	}
	\vspace*{-0.25cm}
	\caption{\label{fig_qe_spectrum}
	Quasi-energy spectrum of the driven  half-BHZ model in an infinitely long ribbon and corresponding asymptotic two terminal differential conductance
	$G_{\text{RL}}^{\infty}(E)$ for  finite system connected to infinite leads. The results are shown for parameters with only trivial gaps with driving amplitude $F=2$
	(Left) and a topological gap at $\epsilon=0$ or $F=8$  (Right).
	The quasi-energy spectra are obtained for strip of width $W=\num{60}$ sites by diagonalization in Sambe space with~\num{5} sidebands.
	The quasi-energies $\varepsilon$  are shown in units of the driving frequency $\hbar \omega$ as a function of the quasi-momentum in the direction
	of the ribbon $k_x$.
	For the topological case ($F = \num{8.}$, Right figure), states appear inside the gap $\epsilon=0$, located on either side of the ribbon.
	The location of each Floquet mode was numerically computed and converted to a color using the color scale of figure \ref{fig:figure_geometry}. Delocalized states (and states located in the bulk) are in green, whereas states located on an edge are in red or blue, depending on which edge. This signals the appearance of a topologically non-trivial gap around $\epsilon=0$. On the other hand, the gaps in
	$\epsilon= \pm \hbar \omega /2$ remains topologically trivial in both cases.

	The asymptotic long time limit of the corresponding two-terminal differential conductance is displayed as a function of  the chemical potential of the left lead
	(black dots and lines) for a finite sample of size $W\times L=\num{60}\times\num{30}$ sites.
	This differential conductance $G_{\text{RL}}^{\infty}$ is quantized, with $G = 1 e^2/h$ in the topological quasi-energy gap (Right) and vanishes in the trivial gap (Left).
	Hence it is an accurate probe of the existence of topological gap in the quasi-energy spectrum of open conductors connected to electrodes.
	%
	The colored arrows indicate the chemical potential of the incoming lead used in the computations of Fig.~\ref{fig_conductance_time_short}.
	}
\end{figure*}
\begin{figure}[hbt]
	\centering
	\pgffig{figure_geometry}

	\caption{\label{fig:figure_geometry} Geometry of the semi-infinite system used to compute quasi-energy spectra of figure \ref{fig_qe_spectrum}. In those spectra, the color of each point represents the average $y$ position of the corresponding Floquet mode, according to the color scale shown on this figure.  }
\end{figure}
Following \cite{Rudner:2013}, we use the restriction to spins up of the Bernevig-Hugues-Zhang model of
quantum spin Hall effect in \ce{HgTe}/\ce{CdTe} quantum wells~\cite{Bernevig:2006} (referred to as half-BHZ model). As the Haldane model~\cite{Haldane:1988} it realizes an anomalous quantum Hall equilibrium phase, but on a square lattice
with  two orbitals per site, denoted $s$ and $p$.
The tight-binding Hamiltonian with nearest and next-to-nearest neighbors hoppings (see Fig.~\ref{fig:reseau})
 can therefore be written as a two-by-two matrix on the $(s,p)$ basis as
\begin{equation}
\begin{split}
	&H_{\text{BHZ}} = \sum_{x,y} \biggl[  \left[\left(M - J - 4 B \right) \sigma_z - \mu \sigma_0 \right] \ket{x,y} \bra{x,y} \\
	&+T^{(1)}_x  \ket{x,y} \bra{x+1,y} +T^{(1)}_y \ket{x,y} \bra{x,y+1} \\
	&+T^{(2)} \left( \ket{x,y}  \bra{x+1,y+1} + \ket{x,y}  \bra{x+1,y-1} \right)	\biggr] \\
	&+ \text{h.c.}
\end{split}
\end{equation}
with hopping matrices
\begin{equation}
\begin{split}
	T^{(1)}_{x/y} = \ \frac{A}{2 \ensuremath{\mathrm{i}}} \sigma_{x/y} + B \sigma_z
	\text{ and }
	T^{(2)} =  \frac{J}{4} \sigma_z
\end{split}
\end{equation}
where $\sigma_{x,y,z}$ are the Pauli matrices and $\sigma_0$ the identity matrix. The parameters of this Hamiltonian are chosen as~$M=\num{-1.}$, $J = \num{1.5}$, $A = \num{4.}$, $B = \num{1.5}$ : they correspond to an equilibrium phase which is a trivial insulator.  A topological insulating phase can be reached by
varying {\it e.g.} $M$ such that $0<M/B<4$ or $4<M/B<8$.
This trivial equilibrium phases is submitted to a periodic on-site perturbation \cite{Rudner:2013}
\begin{equation}
	\Delta H(t) = \sum_{x,y} F \left[  \sin(\omega t) \sigma_x +  \cos(\omega t) \sigma_y \right] \ket{x,y} \bra{x,y}.
\end{equation}
Note that this perturbation does not correspond to the variation of a parameter of the initial Hamiltonian.
Throughout our study, we have used a driving frequency $\omega = \num{20}$.
Depending on the strength $F$,
this perturbation can drive the system either towards a topologically nontrivial out-of-equilibrium state or in a topologically trivial out-of-equilibrium state.
 We have chosen two values $F=2$ and $F=8$ of the driving amplitude so that  gaps open at $\epsilon= 0, \pm \hbar \omega/2$ of the
  quasi-energy spectrum.  For each value of $F$ and each gap,
 we have computed numerically the bulk topological invariant~$W_{\varepsilon}[U]$ associated to the quasi-energy gap around
 $\epsilon$~\cite{Rudner:2013} to ensure that $F=2$ correspond to a trivial gapped Floquet state, while $F=8$ corresponds to a topological  gapped Floquet state
  with a non-trivial gap at $\epsilon= 0$.  Alternatively,  in the two cases
we have computed the quasi-energy spectrum for the driven model in an infinitely long ribbon of width $W=\num{60}$ sites (see Fig.~\ref{fig:figure_geometry})
by diagonalization in Sambe space with~\num{5} sidebands. The resulting spectra are shown in Fig.~\ref{fig_qe_spectrum}: they show as expected that states located at each edge on the ribbon appear
 inside the $\epsilon=0$ gap for the topological case ($F=8$) as opposed to the trivial case~($F=2$).

Finally, to study transport through the system, leads are attached to a finite size system.
These leads are modeled by a simple tight-binding Hamiltonian on a square lattice with nearest neignbors hoppings with amplitude~$J_0=\num{8}$. An onsite potential is added to the lead Hamiltonian to reduce mismatches between the incoming and outgoing states of the leads and the scattering states of the central region.

The numerical calculations are performed using the numerical method described in \cite{Gaury:2013}.
 Although the technique is based on wavefunctions, it is mathematically equivalent to the Green function approach used in this article
(see Eq.~(49) of \cite{Gaury:2013} for the connection with the transmission coefficient as well as section 5.4 for the link with Floquet theory). Our implementation is based on the Kwant package \cite{Groth:2014}.

\begin{figure*}[!ht]
\centerline{
	\pgffig{fig_conductance_time_short_bands}\hspace*{-0.75cm}
	\pgffig{fig_conductance_time_short_gap}
	}
	\vspace*{-0.25cm}
	\caption{\label{fig_conductance_time_short} Two-terminal differential conductance as a function of time, in the transient regime, for two chemical potentials, in a topologically trivial case. Green curve: incoming chemical potential in a quasi-energy band ; blue and red curves: incoming chemical potential in the bulk quasi-energy gap, in a trivial (blue) and a topological (red) case. Inset: longer simulations were carried out to ensure convergence. The asymptotic value is denoted by~$G_{\text{RL}}^{\infty}$.}
\end{figure*}
\subsection{Probing the quasi-energy bands and gaps through two-terminal differential conductance}

The differential conductance $G_{\text{RL}}(t)$ is computed in a two-terminal setup through a sample of width $W=60$ and length $L=\num{30}$ sites.
Throughout this study, the chemical potential of the undriven system was chosen as~$\mu_{\text{sys}}=0$.
To get rid of oscillations faster than the driving frequency, which are irrelevant for our purpose, we perform a sliding average over one driving period~$T$.
After a transient regime, the (averaged) differential conductance~$\bar{G}(t)$ converges to a finite value (see Fig.~\ref{fig_conductance_time_short}).
This transient regime can be understood as the time of flight of the state injected at the left lead to the right lead after the driving perturbation has been turned
on and the Floquet states developed inside the system. When the chemical potential of the incoming lead lies in a topological quasi-energy gap of the scatterng region, transport occurs through a chiral state localized near the edge of the sample.
 We can easily evaluate the travel length $L$. The expected group velocity is
 extracted from the slope of the quasi-energy dispersion relation from Fig.~\ref{fig_qe_spectrum} through
\begin{equation}
	v_{\text{g}} = \frac{1}{\hbar} \frac{\ensuremath{\mathrm{d}} \varepsilon}{\ensuremath{\mathrm{d}} k},
\end{equation}
and we obtain $v_{\text{g}} = \num{0.15(1)} \, a \omega$ where $a$ is the lattice spacing and $\omega$ the driving  frequency. This correlates
perfectly with the time~$\Delta t$ of the first increase from zero of the conductance from the switching on of the driving field. From
 the curve on Fig.~\ref{fig_conductance_time_short} we find  find~$L/\Delta t= \num{0.14(1)} \, a \omega$.

After this transient regime, the differential conductance $G_{\text{RL}}(t)$ reaches a long time stationary limit~$G_{\text{RL}}^{\infty}$, as shown in the inset of
 Fig.~\ref{fig_conductance_time_short}. As expected, this asymptotic differential conductance is
   sensitive to the quasi-energy spectrum of the driven system: when in a spectral band, it reaches high values but
vanishes when the chemical potential $\mu_{\text{L}}- \mu_{\text{sys}}$ lies in a trivial spectral gap of the Floquet operator, as shown on the left plot of
Fig.~\ref{fig_conductance_time_short}. Moreover, when this spectral gap is topological, the associated presence of chiral states at the edge of the system
shown in Fig.~\ref{fig_qe_spectrum} (Right) leads to a quantized two terminal conductance as shown in the right plot of Fig.~\ref{fig_conductance_time_short}.
 To further study the correlation between the differential conductance as a function of the chemical potential $\mu_{\text{L}}- \mu_{\text{sys}}=E$ and
 the spectral gap of the system we have studied the behavior of the long time limit of this conductance as a function of~$E$.  The corresponding results as well the
  spectra of the isolated driven models are plotted in Fig.~\ref{fig_qe_spectrum} for both the trivial and a topological gapped Floquet states.
 We find a strong correlation between the vanishing of both quantities for the topologically trivial case: the differential conductance vanishes only
 inside a trivial gap,  except at the edge of the gap where finite size effects occurs due to imperfect transparencies of the contact with the leads,
 as shown on a magnified view around the gap $\epsilon=0$ in Fig.~\ref{fig_conductance_bias_zoom}. This demonstrates both that the quasi-energy spectrum
 of the finite system connected to infinite leads is sufficiently close to the spectrum of the isolated infinite strip, and that the differential conductance is an accurate
 probe of this spectrum for the open system.
Moreover, in the topological case the asymptotic differential conductance remains constant equal to the number $n=1$ of edge states (in units of~$e^2/h$) inside the topological gap
as shown in Fig.~\ref{fig_conductance_bias_zoom}. The small deviations from $G_{\text{RL}}^{\infty} = 1 \, e^2/h$ visible in figure \ref{fig_conductance_bias_zoom} are attributed to the finite dispersion relation of the leads, which does not completely satisfy the wide band approximation and leads to imperfect transparencies of the contacts.

 %
\begin{figure}[!t]
	\centering
	\pgffig{fig_conductance_bias_zoom}
	\vspace*{-0.5cm}
	\caption{\label{fig_conductance_bias_zoom} Zoom of Fig.~\ref{fig_qe_spectrum} : asymptotic two-terminal differential conductance $G_{\text{RL}}^{\infty}$ in units of $e^{2}/h$
	with respect to the chemical potential of the left lead (black dots and lines)
	around the quasi-energy gap $\epsilon=0$ for a trivial (red) and topological gap (red).
	The reference energy $\mu_{\text{L}}=0$ corresponds to an unbiased lead whose chemical potential identifies with the one of the driven system
	$\mu_{\text{sys}}=0$.
	The differential conductance vanishes in the quasi-energy gaps for a trivial spectral gap, but reaches an approximately constant and quantized value $G =  e^2/h$
	within the topological gap with one chiral edge mode.

	Deviations from $G_{\text{RL}}^{\infty} = 1 \, e^2/h$ are attributed to the finite dispersion relation of the leads, which does not completely satisfy the wide band approximation and leads to imperfect transparencies of the contacts.
	}
\end{figure}

\subsection{Multiterminal geometry}

 A crucial characteristics of the edge states occurring in the topological gap of the spectrum shown in Fig.~\ref{fig_qe_spectrum} (Right)
 is their chiral nature.
 In the equilibrium case this property together with their ballistic propagation leads to quantized conductance in a Hall bar geometry  \cite{Haldane:1988}.
 To test the chirality of the topological edge states, we have computed differential conductances
  in a three-terminal geometry shown in Fig. ~\ref{fig_conductance_time_three_terminal_bands}, where the width of the contact with the electrodes
  is $W=\num{30}$ sites, the total length (between $L$ and $R$ contacts) is $\num{50}$ sites, corresponds to all three arms having a length of $\num{10}$ sites.

 In this geometry, we monitor the two differential conductances
 %
 \begin{equation}
 \begin{split}
     G_{R,T} (E) =  - e \left. \frac{\ensuremath{\mathrm{d}} \mathcal{I}_{R}}{\ensuremath{\mathrm{d}} \mu_{T}}  \right|_{\mu_{T}=\mu_{\text{sys}}+E} , \\
     G_{L,T} (E) =  - e \left. \frac{\ensuremath{\mathrm{d}} \mathcal{I}_{L}}{\ensuremath{\mathrm{d}} \mu_{T}}  \right|_{\mu_{T}=\mu_{\text{sys}}+E} .
\end{split}
 \end{equation}
 where $\mathcal{I}_{R,L}$ are the current in the $R,L$ contacts averaged over one period $T$ of drive (see eq.~\ref{eq:averagedCurrent}).
 The chemical potential of the system is still set to~$\mu_{\text{sys}}=0$ (as if imposed e.g. by a backgate).
\begin{figure}[!ht]
	\pgffig{fig_conductance_time_three_terminal_gap2}
	\vspace*{-0.5cm}
	\caption{\label{fig_conductance_time_three_terminal_bands} Three-terminal differential conductance $G_{\text{LT}}$ (violet) and
	$G_{\text{RT}}$ (orange) as a function of time for a chemical potential of the top lead in a topological bulk gap.
	After an initial transient regime,  asymptotic values for the nonlocal conductances are reached with
	$G_{\text{LT}}^{\infty} = \num{0.0002(1)} \, e^2/h$ and~$G_{\text{RT}}^{\infty} = \num{0.94(1)} \, e^2/h$.
	The simultaneous vanishing of $G_{\text{LT}}^{\infty}$ and quantized $G_{\text{RT}}^{\infty} $ to the two-terminal value
	demonstrates the chiral nature of topological edge states.
	This is further illustrated in the inset by the corresponding probability density $|\psi(t, x)|^2$ in the stationnary regime (here at $t =1000\ T$) which
	is entirely localized around the $T\to R$ edge, demonstrating furthermore the high transparency of the $R$ contact for this edge mode.
	Simulations are done with a width of the contacts $W=\num{30}$ sites and a total length of \num{50} sites.
	}
\end{figure}

We consider the case of a topological gap round~$\epsilon=0$ (see Fig.~\ref{fig_qe_spectrum} (Right)).
First we set the chemical potential of the top lead inside this gap ($\mu_{T}=0$).
The time evolution of the differential conductances are shown in Fig.~\ref{fig_conductance_time_three_terminal_bands}.  After a transient regime, the differential conductances
converge to asymptotic values $G_{\text{LT}}^{\infty} = \num{0.0002(1)} \, e^2/h$ and~$G_{\text{RT}}^{\infty} = \num{0.94(1)} \, e^2/h$. The value of
$G_{\text{LT}}^{\infty}$ is in agreement with the two terminal results, while the vanishing of $G_{\text{RT}}^{\infty}$ is in perfect
agreement with the chiral nature of the edge state moving clockwise for the chosen parameters. This can be contrasted with the
case of a chemical potential $\mu_{T}=0$ set inside a bulk Floquet band, displayed in Fig.~\ref{fig_conductance_time_three_terminal_bands_bulk}.
\begin{figure}[!ht]
	\pgffig{fig_conductance_time_three_terminal_bands2}
	\vspace*{-0.5cm}
	\caption{\label{fig_conductance_time_three_terminal_bands_bulk}
	Three-terminal differential conductance $G_{\text{LT}}$ (violet) and
	$G_{\text{RT}}$ (orange) as a function of time for a chemical potential of the top lead in a quasi-energy band. A longer transient regime is observed, in agreement with
	smaller group velocities than for the edge states, and large asymptotic values for both $G_{\text{LT}}$ and $G_{\text{RT}}$ are reached, confirming the
	non-chiral nature of the corresponding Floquet states.
	The inset shows color map representing the probability density $|\psi(t, x)|^2$ for at $t/T=1800$, demonstrating the delocalized nature of the Floquet states.
	}
\end{figure}
In this case, after a longer transient regime due to slower group velocities, both conductances converge towards large finite values, confirming the absence of chirality for
these bulk states.

Finally, to illustrate the spatial structure of these Floquet states we show in insets of Fig.~\ref{fig_conductance_time_three_terminal_bands} and~\ref{fig_conductance_time_three_terminal_bands_bulk} a color map of the probability density
of the states reached at large times for both the topological gap and inside the bulk bands. The chiral nature of the states injected in the top lead is clearly apparent
and confirms the transport results.

\section{Discussion}

We have studied the differential conductances $G_{\ell' \ell} (E)$ defined in Eq.~\eqref{eq:diff_cond} for both a two-terminal and a three-terminal geometry.
As for equilibrium phases,
we have found that these conductances probe the topological nature of a gapped Floquet state: it
vanishes whenever the chemical potential $\mu_{\ell}$ lies in a topologically trivial gap while we found it to be quantized
for chemical potential  inside a topological gap of quasi-energies. Moreover, the chiral nature of the edge states reflected on the value of the multi-terminal
differential conductances within this gap.
 These results validate the relation between the DC transport through a periodically driven system and the Floquet description of the dynamics of the isolated
 system. This requires that the dissipation occurs mostly inside the leads, {\it i.e.} that the system is small enough for the travel times through it to be small compared with
 times scales of other source of dissipation (phonons, photons, etc). In that situation,
  the differential conductance of the system depends on the intermediate time evolution of the driven system and can be accurately described starting from the unitary
 dynamics of the driven system and its associated topological properties \cite{Rudner:2013,Carpentier:2015}.
 This should be the case as long as phase coherence is preserved on the length scale of the sample, even in the presence of dissipation or weak interactions.
  This result paves the way towards a direct engineering
 of a topological filter through the implementation of Floquet states.

 Our results on the quantization of differential conductances inside a topological gap contrast with some of the recent results on similar systems \cite{Kundu:2013,FoaTorres:2014,Farrell:2015}, where a summation procedure over
 multiple chemical potentials was required to recover a quantized conductance. Note that these previous results used a different measuring scheme of differential
 conductance, symmetric in chemical potentials of electrodes, which may filter out some of the energies of the outgoing electronic states, and thus require
 a summation protocol. In our case, a strong and asymmetric voltage bias is used to compensate the time dependence (through quasi-energies) of the Floquet states.
 Moreover, we have used in our numerical study a model with a single edge mode located around $k\simeq 0$ as opposed to double-mode topological gaps
  used previously.  More complex edge structures may well lead to lower transparency of the interfaces for energies in the gap, and thus non-quantized conductances.
  The study of the transparency of such interface between a strongly driven conductor and a DC electrode is a deep and topical subject whose
  discussion goes beyond the scope of the present study.

{\bf Acknowledgments:} This paper is dedicated to the memory of Markus Büttiker whose pioneering works on mesoscopic systems and in particular on scattering theory of chiral edge channels shaped our understanding of electronic transport. P.D. will always be grateful to him for his trust, enthusiasm and advice he received during his stay as a post-doc assistant in the Büttiker's group.
This work was supported by the French Agence Nationale de la Recherche (ANR) under grants SemiTopo (ANR-12-BS04-0007), IsoTop (ANR-10-BLAN-0419) and TopoDyn (ANR-14-ACHN-0031) and by the European Research Council grant MesoQMC (ERC-2010-StG\_20091028 257241).

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