One of the most important reminders of the slaughter in Orlando and San Bernadino is that terrorists often prefer to attack "soft" targets because they usually lack strict security, and there are many more soft targets than hard targets. This is hardly a new phenomenon, with hotels (in Jakarta and Mombassa), restaurants/night clubs (Bali and Tel Aviv), museums (Tunis), places of worship (Istanbul), trains (Madrid) and buses (France, Israel and Turkey) all having been prime targets in the past. This being the case, greater effort should be made to implement at least minimal security for soft targets that have proven to be particularly appealing to terrorists. If this can be done in developing countries with meager financial resources, it is certainly achievable in the developed world.
In many parts of the world, the entrances to metro rail systems are checked, as are entrances to department stores, office buildings, and shopping centers. Security personnel check everyone's bag or purse as individuals enter. Is this a guarantee that a gun or bomb will not be smuggled onto a train? Of course not, but apart from providing at least some peace of mind, it is a sufficient deterrent to prevent would-be terrorists from attacking with impunity. Had such a system been implemented in many of the above-mentioned examples, most, if not all, of the attacks may have been thwarted, or their severity at least significantly reduced.
This implies a greater effort on the part of businesses and government working in tandem to adequately address the problem, for neither have sufficient resources to do so on their own. Since businesses' and governments' approaches to addressing the issue of security are generally not in sync, all parties need to have incentives and guidelines in place so that they may act in a consistent and effective fashion. Increasing safeguards and enhancing general awareness is a precursor to meaningfully addressing the persistent nature of the threat, but this can have unintended consequences.
For example, for many years following the 9/11 attacks, the U.S. was on 'orange' alert, meaning that an attack could happen at any time, and that the threat was ever present. However, after the passage of time, average citizens began to pay little attention to the threat, particularly given that no attacks had occurred. Similarly, though we are collectively appalled and sensitized to the risks of terrorism each time an incident occurs, most of us believe that the law of averages is on our side, and that the risk of actually becoming a statistic of terrorism is extremely low - and it is. We therefore go about our lives with terrorism perhaps being in the back of our minds, as an unfortunate reality of the 21st century, but usually thinking it will end up being someone else's problem, or not thinking about it at all until the next attack occurs.
That is now changing. Attacks on soft targets are becoming more commonplace, and will become even more so with the passage of time. Just as climate change and cyber risk are no longer someone else's problem, half a world away, the era of global terrorism is upon all of us, and impacts us on a daily basis in ways we could not even have imagined 15 years ago. Security protocols have indeed come a long way since then, but we have a very long way to go before we can say that we really are doing a reasonable job of making ourselves not just feel more secure, but be more secure.
Reducing the possibility that an attack can occur, will be successful, or will cause significant damage by enhancing security protocols goes hand-in-hand with changing perceptions that an attack may occur at all. There is also a risk that constantly increasing security spending may both enhance the cost of an attack and reverse any gains made in changing risk perceptions if an attack occurs and is successful.
Spending the money necessary to reduce vulnerability to attack is perhaps the greatest challenge facing developed country governments and businesses. This is because most developed countries did not construct infrastructure and buildings with the type of security safeguards necessary to thwart terrorist attacks. When power transmission sub-stations, railroad tracks or office buildings were constructed, security was not a foremost consideration. The truth is, most developed countries remain highly vulnerable to attacks on soft targets. One of the ironies in the terrorism landscape is that this same type of concern is not as pronounced in many developing countries, where security was always a primary concern.
Will countries in the developed world that currently do not have security systems and personnel in place in shopping malls, movie theaters, office buildings, and other public places ultimately need to as many developing countries have done? The answer is '"yes." And businesses should be prepared to share the cost of implementing these measures. If governments and businesses are to win the battle against terrorism, they must adopt a completely proactive stance--anticipating likely targets, substantially beefing up security, and establishing countermeasures to prevent future attacks from occurring.
While Western government have, by and large, done a credible job of enhancing their ability to identify terrorists and prevent them from entering countries through airports and ports, they have not done as good a job at creating and implementing effective countermeasures among soft targets. It really is surprising that more attacks among soft targets have not happened in the West and the U.S. since 9/11. Undoubtedly, part of that is due to the intelligence agencies being very good at what they do, but part of it also is that the terrorist have fumbled more than once (the Shoe Bomber and the Underwear Bomber being prime examples). Now that lone wolves have permeated the landscape, the urgency of the issue is growing by the day.
What needs to be done? Border controls and identity checks must become even greater, intelligence sharing across borders must become easier and more comprehensive, and background checks on gun owners needs to become routine. Countries that are unaccustomed to implementing strict security precautions on their citizenry will need to begin doing so. Since it appears likely that the days of being able to casually walk into a shopping center without undergoing a security check are numbered, it would be sensible for business leaders to engage in a discussion with governments about how this is going to be accomplished - now. The sooner this is done, the sooner the general public will be conditioned to accept the inevitable--we face a future when security will saturate many aspects of our daily routines, whether dining in a restaurant, boarding a bus, or entering an office building.
Terrorism has and will continue to affect how business operates, and how much it costs to operate. It has become necessary for businesses to begin to bear the cost of installing security systems and personnel to safeguard their customers and employees on a widespread basis. Businesses and governments must reach an understanding about how the problem of providing meaningful security is going to be addressed in the future. A coordinated, purposeful, definitive approach to creating a more secure environment must be achieved if the West is to stay a step ahead of the terrorists.
It is surely in the mutual interest of business and government that this occurs sooner rather than later. Businesses may as well embrace the reality that providing funding to achieve meaningful security to protect their physical operations, employees, and customers is not only necessary, but also a smart investment. All one needs do is consider the enormous costs implied in enduring political violence, terrorism, or business interruption losses--and the heightened insurance premiums that result from them--to realize that preventive action is prudent, sensible and cost-effective in the long term. More importantly, businesses have a moral obligation to help ensure the safety of their employees and patrons, and businesses are already being held accountable for not enhancing their security protocols. It is time to get out ahead of this issue, for the costs of waiting can only rise.
*Daniel Wagner is CEO of Country Risk Solutions and co-author of the book "Global Risk Agility and Decision Making".


An average price for services of a professional photographer in Prague is 150 euro an hour.
On our website you can find both affordable and high-end photographers, whose services start at 2000 euro for the entire day. To get more detailed information and inquire about a final cost of a photoshoot, simply contact a desired professional specifying the dates and the itinerary of your trip as well as the type of a photoshoot you are interested in.


A contemporary Italian floor lamp found in Italy that showcases a rectangular cage of perforated iron surrounding a single light source. The base of this modern Italian lamp is a square piece of steel reflecting the square outline of the light sheath and that the two sections are connected by a single length of pipe. Please notice that the steel has been allowed to oxidise. The punctured surface of the material surrounding the light source focuses attention on the bulb while casting an unusual pattern of illumination. The extreme simplicity of this design creates a powerful sculptural presence and will be intriguing in a modern interior or will provide an unexpected counterpoint in a more traditionally designed space.
Startling contemporary floor lamp found in Italy that showcases a rectangular cage of perforated iron surrounding a single light source. The base is a square piece of steel reflecting the square outline of the light sheath and that the two sections are connected by a single length of pipe. Please notice that the steel has been allowed to oxidise. The punctured surface of the material surrounding the light source focuses attention on the bulb while casting an unusual pattern of illumination. The extreme simplicity of this design creates a powerful sculptural presence and will be intriguing in a modern interior or will provide an unexpected counterpoint in a more traditionally designed space.


No expense or time was spared on this ground up, nut and bolt, restoration at the factory and the re-paint from Rosso to Rubino is the perfect compromise, and.
making it the only Ferrari and one of a few other cars to have won the French.
The most important Ferrari sports racing car is set to become one of the most.
Ferraris can cost anywhere from about US$200,000 to over US$300,000 per car.
An extra hall has been added to this year's NEC Classic Motor Show – making. Owners' Club and the Piper Sports and Racing Car Club. Andrew Evanson, senior.
Ferrari and Mini Cooper owners' clubs – and individual owners. In addition to the.
8-cylinder V-engine 90 ° with a displacement of 4,308 cc and 510 hp coupled with an F1 shift (6-speed paddle shift). The car has only 11,700 km on the speedometer.
Ferrari. much as a whole other luxury car — you could buy a Mercedes-Benz C-Class ($73,500) or Tesla Model X ($79,500). Ferraris can cost anywhere from.
This time warp Ferrari 328 GTSi is available for a negotiated sale around the.
the Pacific peninsula 90 minutes south of Silicon Valley, for one of the most important classic car shows on earth, the Pebble Beach Concours d'Elegance.
It's a car. classic old-school supercar and easily among the prettiest. In 1976, the 365 morphed into the 512BB which resulted in a hike in displacement, but with.
1991 Repairable Mazda read ad pls 65k mls genuine. Spares or repair only cat n non structural It is with a heavy heart im listing this. Here i have my Mazda 323 SE 1.


text:
All The Photos from Chromeo's Insane DJ Set at Columbus' Otherworld
Electro Funk duo Chromeo played a sold-out DJ set at Columbus' immersive choose-your-own-adventure Sci-Fi art installation Otherworld on Feb. 14.
Photos by Hailey Bollinger
Hailey Bollinger
chromeo,
chromeo columbus,
columbus live music,
otherworld,
otherworld columbus
Everything We Saw During Old Crow Medicine Show's Performance at Taft Theatre
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Billboard Ad

redpajama_set_name: RedPajamaCommonCrawl


No one is quite sure what Google plans to do with all of these pictures in the long run, and it's possible the company hasn't even figured that out. But in a landscape fast becoming dominated by artificial intelligence, data — in this case, your photos — has become its own reward.
Next post Married To Medicine: Great Show That YOU need to Watch!!!


Last week we announced some very big news – Shelton Interactive is being acquired by Advantage Media Group.
I can't tell you how excited I am about this new opportunity, not only because it will open huge doors for our team and our clients/members, but more importantly because I know our companies will positively impact many more lives together than we would apart.
As you can imagine, selling the company wasn't an easy decision and amidst the soul-searching that led to this news I have been thinking a lot about entrepreneurship and the colliding emotions of fear, fun and fulfillment that come with it.
Before I dig into what's ahead for our team at Advantage Media Group, I want to take a look back at a few lessons I learned the hard way as an entrepreneur over the past six years in the hopes that it will help other entrepreneurs and would-be entrepreneurs who are struggling.
Let me start by stating the obvious – starting a business is a very very scary thing to do. In fact, I almost didn't do it – there was a two-year period of time before I started the agency that I talked myself out of it, opting to stay in my comfort zone at my previous position.
Sure, I had wanted to have my own business since I was young, but actually going through with it was a very different thing.
I remember thinking that it felt risky and selfish to make the jump at the time.
I had a good job at a respected book PR agency in Austin and was doing well enough that my wife Paige could stay home with the boys (both still in diapers in 2010), which was a big goal for us when we decided to have kids.
I loved my team at P&P and our clients but I had a nagging feeling that I couldn't shake that a better model could be built.
I remember staring at the ceiling many a night going back and forth on the decision of whether to start a new agency or not.
I knew in my gut that was true but if I'm being honest, I was scared to death to jump out and start the company.
Lesson 1 – Fear is the biggest road block for would-be entrepreneurs and you must be self-aware of that reality. As one of our members, Jon Acuff, says, you have to be willing to "punch fear in the face." That's easier said than done and for a long time I wasn't ready to throw that punch.
There are some good things about fear. It can prevent us from doing stupid things and, even as I look back now, I think I had good reason to be scared of what would happen if it didn't work.
My biggest fear was knowing that if I started the business and it failed, the boys were headed to day care because both Paige and I were going back to work. If that happened I was also afraid of having to ask my Dad for money to bridge me (and Paige) to a new job – not because he wouldn't be there for me (he would have been and was a big mentor for me as I thought through the business) but because I always promised myself I'd never put myself in a position to need to make that ask.
So I talked myself out of it month after month.
We had less than $8,000 in the bank, which isn't much of a runway.
We had a new mortgage, college funds, car payments and diapers to buy.
We were comfortable and I didn't want to rock the boat.
If I'm being totally honest, I was also afraid I couldn't do it – that I wasn't ready from a leadership perspective.
Thankfully, Paige kept pushing me and that nagging feeling I had that a much better agency model was needed wouldn't go away. I was finally pushed over the finish line on the decision while sitting in the back of the room at a session at SXSW Interactive 2010. I don't even remember which session I was in but I remember finally saying "I can do this" and walking out of the room with the decision made.
Lesson 2 – Often times wisdom, insight and encouragement from others will be the thing that gives you the confidence to finally get you over the edge and start your business. Surround yourself with people who can provide that support for you.
A week later I sat down with the owners of the agency where I worked and told them my plan, giving them three months notice based on the prominence of the managing director role I was in and the financial reality of needing to have a bit of a runway to save more money.
I remember being scared to death they were going fire me on the spot.
Thankfully they didn't and I began planning for the agency launch, wrapping up my last day at "work" June 30, 2010.
On July 1, 2010 Shelton Interactive was launched to zero fanfare or media attention. I'm not sure anyone noticed outside of my family, but I was excited to finally be wheels up.
I'd love to tell you that it was a success from the start but those first two months I was a case study for how not to start a business well.
In the first month of business I was defrauded of $4,000 of the first $6,000 in the corporate bank account. It was totally my fault and I remember feeling like such a failure for letting it happen. That was so much money to us at the time and put the agency and my family in risky financial territory.
As if that wasn't enough, each of my first two part-time employees quit within three weeks of me starting the business, one having the courtesy to call to give me the news but not any notice ("Today's my last day") and one quit without even communicating with me at all, taking boxes and boxes of client books with him and never returning calls or giving the books back. To this day I haven't seen or heard from him.
I think neither thought anything was going to come of "Shelton Interactive" and I had a hard time arguing at the time.
Lesson 3 – Be prepared for everything to go wrong in the first few months after you start the business and use it as fuel for the long hours needed to be successful.
Our first two clients were Chicken Soup for the Soul (who came with me from my previous agency) and Dr. Richard Senelick (who wrote the very first check made out to Shelton Interactive) and both were incredibly patient through those rocky first few months.
Cue huge sigh of relief.
We had no money for an office, so I worked from a coffee shop on Parmer Lane near our house called "It's a Grind." I quickly realized that coffee grinders roaring in the background on conference calls doesn't inspire a lot of confidence from clients in a new business.
I was working 60-70 hour weeks (nights, weekends, etc. – which I knew would be part of it) and Paige had to step up in so many ways with the boys. It was very hard on her and boy did she step up.
Lesson 4 – Entrepreneurship is so much easier when you are on the same page with your spouse or partner and can attack it as a team. I can't imagine how hard it would be to build a business without that alignment – I would have failed.
What was counter-intuitive for me is that the early challenges didn't discourage me – they fueled a competitive fire to hustle more to prove those who left wrong and those who stuck with me right. I knew we'd pull out of it and we soon did, thanks to help from friends.
Dr. Julie Silver, at the time head of Harvard Health Publications, and my friend and publishing industry stand-out, Lisa Tener, went out of their way to send me many of my first referrals. Two mentors, Clint Greenleaf and Robbie Vorhaus, each successful entrepreneurs, gave me the benefit of their experience, talking me off the ledge numerous times and helping me build a better business. Heather Adams, a dear old friend and rockstar publicist and entrepreneur in her own right, gave me an open door with Zondervan to work on a book called One Thousand Gifts from a remarkable debut author out of Canada named Ann Voskamp. That campaign was not only was a blessing to us in terms of the relationship and impact but also opened the floodgates for us in terms of interest from Christian publishers. Nick Alter and his team at Alter Endeavors pitched in and helped us on numerous projects. But my biggest external helping hand came from my co-author Barbara Cave Henricks, who gave me a chance before we had any reputation at all, forming a strategic partnership with the agency and giving us a huge shot in the arm of credibility.
We were off to the races, not due to what I had done (screwing up was the only thing I had done well in those early months) – it was thanks to the help of good friends and, most importantly, an awesome team.
Huge early contributions were made by the first members of the Shelton Interactive crew, including Christy McFerren, Amber McGinty, Beth Gwazdosky, David Luna, Will Ruff, Travis Wilson and Taylor and Tiffany Ballard, but we really saw lift off when we added an old colleague and great friend mine, Shelby Sledge, to the team in 2011. Her fingerprints are all over the work, culture and growth of Shelton Interactive and she deserves an immense amount of credit for the agency's success. Shelby's just the best.
Since then, we have had out-sized contributions from our wonderful family of team members, including Vanessa Navarro, Andrea Sanchez, Katie Schnack, Shelby Janner, Tiffany Ballard, Patti Conrad, Brandon Procell, Anthony Aguilar, Sara Pence, Wes Fang, Melanie Cloth, Paige Velasquez, Paige Dillon, Tiffany Jones, Taylor Ballard, Katrina Barber, Whitney "The Wolf" Burnett, Lauren King, Allison Bright, Richard Ricondo, Will Ruff, Sam Joseph, Marshall Weber, Jeremy Strom and others. They have been the lifeblood of our culture and our business and are the real reason for our success.
Lesson 5 – As a young company, it's impossible to hang onto great talent if you don't commit to making great culture the central foundation for the business. As Peter Drucker says, "Culture eats strategy for lunch" and every bit of attention we paid to our culture paid off in spades.
This team helped us land at #15 on the Austin Business Journal's Fast 50 List in 2014, which tracks the fastest growing companies in Central Texas, helped us launch more than 30 New York Times and Wall Street Journal bestsellers and be named by Clutch as one of the top 10 social media marketing agencies in the country in both 2015 and 2016.
Our family of clients – each and every one – also deserve huge credit for helping us get here, as the business has been built entirely on the back of referrals. You've taught me so much and I'm forever grateful to you for being part of the SI family.
It's surreal to sit back and think through all that was put in by so many to get us here to the point where this agency, which I almost talked myself out of starting, came to be one of the greatest joys of my life.
So why would I sell the company that has been such a blessing to me and so many others?
As I mentioned, it wasn't an easy decision, but I believe that joining the Advantage Family will open doors to bigger opportunities for our team, our clients and myself and allow us to make a bigger impact on the world in the process.
When I connected with Advantage Media Group CEO Adam Witty last year, he and I quickly hit it off. He's a YPO'er and a fellow EO member and one of the few people I've met who is as passionate about entrepreneurship, culture and college sports as I am.
After striking up a conversation at an EO event, he said he had heard good things about Shelton Interactive and wanted to explore a potential strategic partnership. I had also heard very good things about Advantage, which has won multiple "best places to work in South Carolina" awards and serves 850 authors in 40 US states and 13 countries. Advantage has grown to become one of the largest business book publishers in America and is famous for its renowned "Talk Your Book" program.
I had turned down many acquisition offers in the past but the more I got to know Adam and the Advantage Family, the more I thought that our companies were kindred spirits. We both put our team first, growing with culture at the heart of our philosophy. We both value providing our members with the best service experience they will ever have. We both want to make a positive impact on the world and, in my judgment, we can do that much better together than apart.
That's the reason I'm so excited about the future. As this chapter closes, an exciting one begins. I'm honored to be joining the Advantage Family, as is our team. We're ready for what the future holds and I'm thankful for lessons learned along the way.


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Home Police: 3 injured in shooting on 38th and Bulloch St.
Police: 3 injured in shooting on 38th and Bulloch St.
Alabama center sells the unclaimed luggage of thousands of airline travelers
SAVANNAH, Ga () — The Savannah Police Department is on the scene of a shooting that took place at 38th and Bulloch Street.
According to police, officers arrived on the scene of a shooting just before 6:00 p.m. on November 4. Police say there were three adult victims in total. Two victims suffered non-life-threatening injuries and one received life-threatening injuries.
VSP: At least 1 injured after fiery crash involving tractor-trailers in Smyth Co.
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UPDATE: POSTED 1/31/2023, 4:36 P.M. The individual killed in the shooting has…
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Additional charges added to man arrested Sunday for burglary and aggravated assault
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The Point, Jan. 31, 2023: Stranded orca buried in Gainesville
Subscribe to The Point, arriving in your inbox Monday through Friday at 8…
New Yorkers become fed up with paying high rent next to migrants' tent city outside Manhattan
Deputy principal Damian Scott Wanstall says he was drunk after trying to pay for sex with girl, 14
Rep. Katie Porter talks about her Senate bid
SARAH VINE: Britain is now a nation split between silent strivers and noisy strikers
Group wedding to be held in St. Augustine on Valentine's Day
New Yorkers are outraged after asylum seekers refused to move from a…
A Sydney deputy principal seeking 'sexy play' from teenagers claims he was…
Rep. Katie Porter talks about her Senate bid – CBS News Watch…
Don't know about you, but I've come to view these endless strikes…


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Where collaboration is the prevailing approach with an abundance of partnerships and an environment promoting cooperation and relationship building.
The Community Network supports individuals and agencies through leadership in the Maple Ridge, Pitt Meadows and Katzie Community. The Community Network responds with commitment, determination and resolve when addressing community needs. The Community Network strongly believes in building the capacity of emerging community leaders through its practice of mentorship and networking. With strength and focus, the Community Network shares the community vision and advocates on behalf of the residents of Maple Ridge, Pitt Meadows, and Katzie.
The Community Network works together in a cooperative manner. The Community Network strives to build connections of trust and mutual value amongst Community Network members and the community at large. Each achievement is a cause for celebration for all Community Network members.
The Community Network embraces equality and inclusiveness in all its community interactions. Respect and integrity guide our relationships with each other and with the community as a whole. The Community Network acknowledges and appreciates the rights of network partners to make choices that fit their individual or organizational values.
The Community Network shares information in a transparent and open manner. There is effort to establish common language and understanding that supports common goals and outcomes. By actively promoting communication, the Community Network creates a system that values the input of all members, partners and the community as a whole. These systems help provide a context that is consistent and meaningful for new members, emerging leaders and the community as a whole.


"I am very excited to be racing with Team Pelfrey over the upcoming season! While I can't compete full time due to my college commitments, I believe that the races I can participate in will be a great experience." Czuba also added "My goal this season is to be able to race in the lead pack at every event this year and gain a few podiums and even a win along the way."
The 20-year-old from Hollis, New Hampshire will enter his fifth season in an F1600 car, alongside his new teammate, Josh Green. Last season, Czuba finished the F1600 Championship Series in 4th place among 32 competitors. This included two 1st place and a 2nd place finish in U.S. Major Tour events as well as a 2nd place finish and multiple top five finishes in the F1600 Championship Series.
Dexter Czuba has quite the impressive bio. Besides working to obtain a degree in Mechanical Engineering at Worcester Polytechnic Institute (WPI), he enjoys and has excelled in skiing, tennis, astronomy, and robotics.
Czuba's first event with Team Pelfrey will be Hoosier racing tires SCCA Super Tour at Virginia International Raceway in April. With the announcement of Czuba, Team Pelfrey is well on its way back to a three, and possibly four car line up and plans to be a dominating force throughout the year.


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Time: Every Thursday from 5 pm till late.


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Tensor product
Tensor product explained
In mathematics, the tensor product
V ⊗ W
of two vector spaces and (over the same field) is a vector space which can be thought of as the space of all tensors that can be built from vectors from its constituent spaces using an additional operation which can be considered as a generalization and abstraction of the outer product. Because of the connection with tensors, which are the elements of a tensor product, tensor products find uses in many areas of application including in physics and engineering, though the full theoretical mechanics of them described below may not be commonly cited there. For example, in general relativity, the gravitational field is described through the metric tensor, which is a field (in the sense of physics) of tensors, one at each point in the space-time manifold, and each of which lives in the tensor self-product of tangent spaces
at its point of residence on the manifold (such a collection of tensor products attached to another space is called a tensor bundle).
Tensors in finite dimensions, and the outer product
The concept of tensor product generalizes the idea of forming tensors from vectors using the outer product, which is an operation that can be defined in finite-dimensional vector spaces using matrices: given two vectors
v\inV
w\inW
written in terms of components, i.e.
\mathbf = \begin v_1 \\ v_2 \\ \cdots \\ v_n \end
\mathbf = \begin w_1 \\ w_2 \\ \cdots \\ w_m \end
their outer product or is given by
\mathbf \otimes \mathbf = \begin v_1 w_1 && v_1 w_2 && \cdots && v_1 w_m \\ v_2 w_1 && v_2 w_2 && \cdots && v_2 w_m \\ \cdots \\ v_n w_1 && v_n w_2 && \cdots && v_n w_m\end
or, in terms of elements, the
-th component is
(\mathbf \otimes \mathbf)_ = v_i w_j .
The matrix formed this way corresponds naturally to a tensor
, where such is understood as a multilinear functional, by sandwiching it with matrix multiplication between a vector and its dual, or transpose:
T(\mathbf^T, \mathbf^T) := \mathbf^T (\mathbf \otimes \mathbf) \mathbf
It is important to note that the tensor, as written, takes two vectors - this is an important point that will be dealt with later. In the case of finite dimensions, there is not a strong distinction between a space and its dual, however, it does matter in infinite dimensions and, moreover, getting the regular-vs-dual part right is essential to ensuring that the idea of tensors being developed here corresponds correctly to other senses in which they are viewed, such as in terms of transformations, which is common in physics.
The tensors constructed this way generate a vector space themselves when we add and scale them in the natural componentwise fashion and, in fact, multilinear functionals of the type given can be written as some sum of outer products, which we may call pure tensors or simple tensors. This is sufficient to define the tensor product when we can write vectors and transformations in terms of matrices, however, to get a fully general operation, a more abstract approach will be required. Especially, we would like to isolate the "essential features" of the tensor product without having to specify a particular basis for its construction, and that is what we will do in the following sections.
Abstracting the tensor product
To achieve that aim, the most natural way to proceed is to try and isolate an essential characterizing property which will describe, out of all possible vector spaces we could build from V and W, the one which (up to isomorphism) is their tensor product, and which will apply without consideration of any arbitrary choices such as a choice of basis. And the way to do this is to flip the tensor concept "inside out" - instead of viewing the tensors as an object which vectors in the manner of a bilinear map, we will view them instead as objects to a bilinear map. The trick is in recognizing that the Kronecker product "" regarding which vectors went into it: the ratios of vector components can be derived from
\frac = \frac = \frac
and from those ratios, the individual components themselves recovered. As a result, a single Kronecker outer product can be used in lieu of the pair
(v,w)
of vectors that formed it, and conversely. Most importantly, this means we can write any map
f:V x W\toZ,
for any third vector space Z, as a map
f\operatorname{T
} : V \otimes W \to Z where
f_T(\mathbf \otimes \mathbf) := f(\mathbf, \mathbf)
The, then, is that if we have the combining operation
⊗ ,
and we are given any bilinear map of the form mentioned, there is such
} that meets this requirement. This is not hard to see if we expand in terms of bases, but the more important point of it is that it can be used as a way to characterize the tensor product - that is, we can use it to the tensor product axiomatically with no reference to such. However, before we can do that, we first need to show that the tensor product exists and is unique for vector spaces V and W and, to do that, we need a construction.
The constructive tensor product
The free vector space
To perform such a construction, the first step we will consider involves introducing something called a "free vector space" over a given set. The thrust behind this idea basically consists of what we said in the first section above: since a generic tensor
can be written by the double sum
T = \sum_^ \sum_^ (v_i w_j) (\mathbf_i \otimes \mathbf_j)
the most natural way to approach this problem is somehow to figure out how we can "forget" about the specific choice of bases
that are used here. In mathematics, the way we "forget" about representational details of something is to establish an identification that tells us that two different things that are to be considered representations of the same thing are in fact such, i.e. which, given those says either "yes, they are" or "no, they aren't", and then "lump together" all representations as constituting the "thing represented" without reference to any one in particular by packaging them all together into a single set. In formal terms, we first build an equivalence relation, and then take the quotient set by that relation.
But before we can do that, we first need to develop what we are going to take the equivalence relation over. The way we do that is to approach this the other way around, from the "bottom up": since we are not guaranteed a, at least constructible, basis when starting from arbitrary vector spaces, we might instead try to start by guaranteeing we have one—that is, we will start first by considering a "basis", on its own, as given, and then building the vector space on top. To that end, we accomplish the following: suppose that
is some set, which we could call an abstract basis set. Now consider all formal expressions of the form
\mathbf = a_1 \beta_1 + a_2 \beta_2 + \cdots + a_n \beta_n
of arbitrary, but finite, length
and for which
are scalars and
\betaj
are members of
Intuitively, this is a linear combination of the basis vectors in the usual sense of expanding an element of a vector space. We call this a "formal expression" because technically it is illegal to multiply
aj\betaj
since there is no defined multiplication operation by default on an arbitrary set and arbitrary field of scalars. Instead, we will "pretend" (similar to defining the imaginary numbers) that this refers to something, and then will go about manipulating it according to the rules we expect for a vector space, e.g. the sum of two such strings using the same sequence of members of
(a_1 \beta_1 + a_2 \beta_2 + \cdots + a_n \beta_n) + (b_1 \beta_1 + b_2 \beta_2 + \cdots + b_n \beta_n) = (a_1 + b_1) \beta_1 + (a_2 + b_2) \beta_2 + \cdots + (a_n + b_n) \beta_n
where we have used the associative, commutative, and distributive laws to rearrange the first sum into the second. Continuing this way for scalar multiples and all different-length combinations of vectors allows us to build up a vector addition and scalar multiplication on this set of formal expressions, and we call it the free vector space over
F(B).
Note that the elements of
considered as length-one formal expressions with coefficient 1 out front, form a Hamel basis for this space.
The tensor product expression is then abstracted by considering that if
\gammaj
represent "abstract basis vectors" from two sets
i.e. that "
\betaj=ej
" and "
\gammaj=fj
", then pairs of these in the Cartesian product
B x G,
(\betai,\gammaj)
are taken as standing for the tensor products
ei ⊗ fj.
(Note that the tensor products in the expression are in some sense "atomic", i.e. additions and scalar multiplications do not split them up into anything else, so we can replace them with something different without altering the mathematical structure.) With such an identification, we can thus define the tensor product of two free vector spaces
F(B)
F(G)
as being something (yet to be decided) that is isomorphic to
F(B x G).
The equivalence relation
The above definition will work for any vector space in which we specify a basis, since we can just rebuild it as the free vector space over that basis: the above construction exactly mirrors how you represent vectors via the Hamel basis construction by design. In effect, we haven't gained anything ... until we do this.
Now, we are not assuming access to bases for vector spaces
that we want to form the tensor product
of. Instead, we will take of
as "basis" to build up the tensors. This is the next best thing and the one thing we are to be able to do, regardless of any concerns in finding a specific basis; this corresponds to adding together arbitrary outer products
of arbitrary vectors. The only difference here is that if we use the free vector space construction and form the obvious
F(V) ⊗ F(W)=F(V x W),
it will have many redundant versions of what should be the same tensor; going back to our basisful case if we consider the example where
V=W=\R2
in the standard basis, we may consider that the tensor formed by the vectors
x=\begin{bmatrix}0&3\end{bmatrix}T
y=\begin{bmatrix}5&-3\end{bmatrix}T,
T := \mathbf \otimes \mathbf = \begin 0 & 0 \\ 15 & -9 \end,
could be represented by other sums, such as the sum using individual basic tensors
ei ⊗ ej,
T = 0\left(\mathbf_1 \otimes \mathbf_1\right) + 0\left(\mathbf_1 \otimes \mathbf_2\right) + 15\left(\mathbf_2 \otimes \mathbf_1\right) - 9\left(\mathbf_2 \otimes \mathbf_2\right).
These, while equal expressions in the concrete case, would correspond to distinct elements of the free vector space
F(V x W),
T = (x, y)
in the first case and
T = 0(e_1, e_1) + 0(e_1, e_2) + 15(e_2, e_1) - 9(e_2, e_2)
in the second case. Thus we must condense them—this is where the equivalence relation comes into play. The trick to building it is to note that given any vector
in a vector space, it is always possible to represent it as the sum of two other vectors
not equal to the original. If nothing else, let
be any vector and then take
b:=x-a
—which also shows that if we are given one vector and then a second vector, we can write the first vector in terms of the second together with a suitable third vector (indeed in many ways—just consider scalar multiples of the second vector in the same subtraction.).
This is useful to us because the outer product satisfies the following linearity properties, which can be proven by simple algebra on the corresponding matrix expressions:
\begin (\mathbf \otimes \mathbf)^\mathsf &= (\mathbf \otimes \mathbf) \\ (\mathbf + \mathbf) \otimes \mathbf &= \mathbf \otimes \mathbf + \mathbf \otimes \mathbf \\ \mathbf \otimes (\mathbf + \mathbf) &= \mathbf \otimes \mathbf + \mathbf \otimes \mathbf \\ c (\mathbf \otimes \mathbf) &= (c\mathbf) \otimes \mathbf = \mathbf \otimes (c\mathbf)\end
If we want to relate the outer product
to, say,
⊗ w,
we can use the first relation above together with a suitable expression of
as a sum of some vector and some scalar multiple of
e1.
Equality between two concrete tensors is then obtained if using the above rules will permit us to rearrange one sum of outer products into the other by suitably decomposing vectors—regardless of if we have a set of actual basis vectors. Applying that to our example above, we see that of course we have
\begin \mathbf &= 0 \mathbf_1 + 3 \mathbf_2 \\ \mathbf &= 5 \mathbf_1 - 3 \mathbf_2\end
for which substitution in
T = \mathbf \otimes \mathbf
T = \left(0 \mathbf_1 + 3 \mathbf_2) \otimes (5 \mathbf_1 - 3 \mathbf_2\right)
and judicious use of the distributivity properties lets us rearrange to the desired form. Likewise, there is a corresponding "mirror" manipulation in terms of the free vector space elements
(x,y)
(e1,e1),
etc., and this finally leads us to the formal definition of the tensor product.
Putting all the construction together
The abstract tensor product of two vector spaces
over a common base field
is the quotient vector space
V \otimes W := F(V \times W)/
\sim
is the equivalence relation of generated by assuming that, for each
(v',w')
taken as formal expressions in the free vector space
the following hold:
Identity.
(v,w)\sim(v,w).
Symmetry.
(v,w)\sim(v',w')
implies
(v',w')\sim(v,w).
Transitivity.
(v',w')\sim(v'',w'')
(v,w)\sim(v'',w'').
Distributivity.
(v,w)+(v',w)\sim(v+v',w)
(v,w)+(v,w')\sim(v,w+w').
Scalar multiples.
c(v,w)\sim(v,cw)
c(v,w)\sim(cv,w).
and then testing equivalence of generic formal expressions through suitable manipulations based thereupon. Arithmetic is defined on the tensor product by choosing representative elements, applying the arithmetical rules, and finally taking the equivalence class. Moreover, given any two vectors
w\inW,
the equivalence class
[(v,w)]
is denoted
v ⊗ w.
Elements of
are often referred to as tensors, although this term refers to many other related concepts as well.[1] If belongs to and belongs to, then the equivalence class of is denoted by
v ⊗ w,
which is called the tensor product of with . In physics and engineering, this use of the
symbol refers specifically to the outer product operation; the result of the outer product
is one of the standard ways of representing the equivalence class
[2] An element of
that can be written in the form
is called a or . In general, an element of the tensor product space is not a pure tensor, but rather a finite linear combination of pure tensors. For example, if
are linearly independent, and
are also linearly independent, then
v1 ⊗ w1+v2 ⊗ w2
cannot be written as a pure tensor. The number of simple tensors required to express an element of a tensor product is called the tensor rank (not to be confused with tensor order, which is the number of spaces one has taken the product of, in this case 2; in notation, the number of indices), and for linear operators or matrices, thought of as tensors (elements of the space
V ⊗ V*
), it agrees with matrix rank.
Given bases
\left\{vi\right\}
\left\{wj\right\}
for and respectively, the tensors
\left\{vi ⊗ wj\right\}
form a basis for
Therefore, if and are finite-dimensional, the dimension of the tensor product is the product of dimensions of the original spaces; for instance
\Rm ⊗ \Rn
is isomorphic to
\Rmn.
Tensor product of linear maps
The tensor product also operates on linear maps between vector spaces. Specifically, given two linear maps
S:V\toX
T:W\toY
between vector spaces, the and is a linear mapS \otimes T : V \otimes W \to X \otimes Ydefined by(S \otimes T)(v \otimes w) = S(v) \otimes T(w).
In this way, the tensor product becomes a bifunctor from the category of vector spaces to itself, covariant in both arguments.[3]
If and are both injective, surjective or (in the case that,,, and are normed vector spaces or topological vector spaces) continuous, then
S ⊗ T
is injective, surjective or continuous, respectively.
By choosing bases of all vector spaces involved, the linear maps and can be represented by matrices. Then, depending on how the tensor
is vectorized, the matrix describing the tensor product
is the Kronecker product of the two matrices. For example, if, and above are all two-dimensional and bases have been fixed for all of them, and and are given by the matricesA=\begin a_ & a_ \\ a_ & a_ \\ \end, \qquad B=\begin b_ & b_ \\ b_ & b_ \\ \end,
respectively, then the tensor product of these two matrices is
\begin a_ & a_ \\ a_ & a_ \\ \end \otimes \begin b_ & b_ \\ b_ & b_ \\ \end = \begin a_ \begin b_ & b_ \\ b_ & b_ \\ \end & a_ \begin b_ & b_ \\ b_ & b_ \\ \end \\[3pt] a_ \begin b_ & b_ \\ b_ & b_ \\ \end & a_ \begin b_ & b_ \\ b_ & b_ \\ \end \\ \end = \begin a_ b_ & a_ b_ & a_ b_ & a_ b_ \\ a_ b_ & a_ b_ & a_ b_ & a_ b_ \\ a_ b_ & a_ b_ & a_ b_ & a_ b_ \\ a_ b_ & a_ b_ & a_ b_ & a_ b_ \\ \end.
The resultant rank is at most 4, and thus the resultant dimension is 4. Note that here denotes the tensor rank i.e. the number of requisite indices (while the matrix rank counts the number of degrees of freedom in the resulting array). Note
\operatorname{Tr}A ⊗ B=\operatorname{Tr}A x \operatorname{Tr}B.
A dyadic product is the special case of the tensor product between two vectors of the same dimension.
Universal property
In the context of vector spaces, the tensor product
and the associated bilinear map
\varphi:V x W\toV ⊗ W
are characterized up to isomorphism by a universal property regarding bilinear maps. (Recall that a bilinear map is a function that is separately linear in each of its arguments.) Informally,
\varphi
is the most general bilinear map out of
V x W.
This characterization can simplify proofs about the tensor product. For example, the tensor product is symmetric, meaning there is a canonical isomorphism:V \otimes W \cong W \otimes V.To construct, say, a map from
W ⊗ V,
it suffices to give a bilinear map
h:V x W\toW ⊗ V
that maps
w ⊗ v.
Then the universal property of
factors into a map
\tilde{h}:V ⊗ W\toW ⊗ V.
A map
\tilde{g}:W ⊗ V\toV ⊗ W
in the opposite direction is similarly defined, and one checks that the two linear maps
\tilde{h}
\tilde{g}
are inverse to one another by again using their universal properties.
The universal property is extremely useful in showing that a map to a tensor product is injective. For example, suppose we want to show that
\R ⊗ \R
\R.
Since all simple tensors are of the form
a ⊗ b=(ab) ⊗ 1,
and hence all elements of the tensor product are of the form
x ⊗ 1
by additivity in the first coordinate, we have a natural candidate for an isomorphism
\R → \R ⊗ \R
given by mapping
x ⊗ 1,
and this map is trivially surjective.
Showing injectivity directly would involve somehow showing that there are no non-trivial relationships between
y ⊗ 1
x ≠ y,
which seems daunting. However, we know that there is a bilinear map
\R x \R → \R
given by multiplying the coordinates together, and the universal property of the tensor product then furnishes a map of vector spaces
\R ⊗ \R → \R
which maps
x,
and hence is an inverse of the previously constructed homomorphism, immediately implying the desired result. Note that, a priori, it is not even clear that this inverse map is well-defined, but the universal property and associated bilinear map together imply this is the case.
Similar reasoning can be used to show that the tensor product is associative, that is, there are natural isomorphismsV_1 \otimes \left(V_2\otimes V_3\right) \cong \left(V_1\otimes V_2\right) \otimes V_3.Therefore, it is customary to omit the parentheses and write
V1 ⊗ V2 ⊗ V3
, so the ijk-th component of
U ⊗ V ⊗ W
(u ⊗ v ⊗ w)ijk=uivjwk
similar to the first example on this page.
The category of vector spaces with tensor product is an example of a symmetric monoidal category.
The universal-property definition of a tensor product is valid in more categories than just the category of vector spaces. Instead of using multilinear (bilinear) maps, the general tensor product definition uses multimorphisms.[4]
Tensor powers and braiding
Let be a non-negative integer. The th tensor power of the vector space is the -fold tensor product of with itself. That isV^ \;\overset\; \underbrace_.
\sigma
of the set
\{1,2,\ldots,n\}
determines a mapping of the th Cartesian power of as follows:
\begin \sigma : V^n \to V^n \\ \sigma\left(v_1,v_2,\ldots,v_n\right) = \left(v_, v_,\ldots,v_\right)\end
\varphi : V^n \to V^
be the natural multilinear embedding of the Cartesian power of into the tensor power of . Then, by the universal property, there is a unique isomorphism
\tau_\sigma : V^ \to V^
such that
\varphi\circ\sigma = \tau_\sigma\circ\varphi.
The isomorphism
\tau\sigma
is called the braiding map associated to the permutation
\sigma.
Product of tensors
See also: Classical treatment of tensors. For non-negative integers and a type
(r,s)
tensor on a vector space is an element of
T^r_s(V) = \underbrace_r \otimes \underbrace_s = V^ \otimes \left(V^*\right)^.
V*
is the dual vector space (which consists of all linear maps from to the ground field).
There is a product map, called the T^r_s (V) \otimes_K T^_ (V) \to T^_(V).
It is defined by grouping all occurring "factors" together: writing
for an element of and
for an element of the dual space,(v_1 \otimes f_1) \otimes (v'_1) = v_1 \otimes v'_1 \otimes f_1.
Picking a basis of and the corresponding dual basis of
naturally induces a basis for
r(V)
(this basis is described in the article on Kronecker products). In terms of these bases, the components of a (tensor) product of two (or more) tensors can be computed. For example, if and are two covariant tensors of orders and respectively (i.e.
F\in
G\in
), then the components of their tensor product are given by[5] (F \otimes G)_ = F_ G_.
Thus, the components of the tensor product of two tensors are the ordinary product of the components of each tensor. Another example: let be a tensor of type with components
\alpha
\beta
and let be a tensor of type
with components
V\gamma.
Then\left(U \otimes V\right)^\alpha _\beta ^\gamma = U^\alpha _\beta V^\gamma
and(V \otimes U)^ _\sigma = V^\mu U^\nu _\sigma.
Tensors equipped with their product operation form an algebra, called the tensor algebra.
Evaluation map and tensor contraction
For tensors of type there is a canonical evaluation map
V \otimes V^* \to K
defined by its action on pure tensors:
v \otimes f \mapsto f(v).
More generally, for tensors of type
(r,s),
with, there is a map, called tensor contraction,
T^r_s (V) \to T^_(V).
(The copies of
on which this map is to be applied must be specified.)
On the other hand, if
is, there is a canonical map in the other direction (called the coevaluation map)
\begin K \to V \otimes V^* \\ \lambda \mapsto \sum_i \lambda v_i \otimes v^*_i \end
v1,\ldots,vn
is any basis of
is its dual basis. This map does not depend on the choice of basis.[6]
The interplay of evaluation and coevaluation can be used to characterize finite-dimensional vector spaces without referring to bases.[7]
Adjoint representation
The tensor product
s(V)
may be naturally viewed as a module for the Lie algebra
End(V)
by means of the diagonal action: for simplicity let us assume
r=s=1,
then, for each
u\inEnd(V),
u(a \otimes b) = u(a) \otimes b - a \otimes u^*(b),
u*\inEnd\left(V*\right)
is the transpose of, that is, in terms of the obvious pairing on
V ⊗ V*,
\langle u(a), b \rangle = \langle a, u^*(b) \rangle.
There is a canonical isomorphism
1(V)
\toEnd(V)
given by
(a \otimes b)(x) = \langle x, b \rangle a.
Under this isomorphism, every in
may be first viewed as an endomorphism of
and then viewed as an endomorphism of
End(V).
In fact it is the adjoint representation of
Relation of tensor product to Hom
Given two finite dimensional vector spaces, over the same field, denote the dual space of as, and the -vector space of all linear maps from to as . There is an isomorphism,
U^* \otimes V \cong \mathrm(U, V),
defined by an action of the pure tensor
f ⊗ v\inU* ⊗ V
on an element of
(f \otimes v)(u) = f(u) v.
Its "inverse" can be defined using a basis
\{ui\}
and its dual basis
\{u
i\}
as in the section "Evaluation map and tensor contraction" above:
\begin \mathrm (U,V) \to U^* \otimes V \\ F \mapsto \sum_i u^*_i \otimes F(u_i). \end
This result implies
\dim(U \otimes V) = \dim(U)\dim(V),
which automatically gives the important fact that
\{ui ⊗ vj\}
forms a basis for
U ⊗ V
\{ui\},\{vj\}
are bases of and .
Furthermore, given three vector spaces,, the tensor product is linked to the vector space of all linear maps, as follows:\mathrm (U \otimes V, W) \cong \mathrm (U, \mathrm(V, W)).This is an example of adjoint functors: the tensor product is "left adjoint" to Hom.
Tensor products of modules over a ring
See main article: Tensor product of modules. The tensor product of two modules and over a commutative ring is defined in exactly the same way as the tensor product of vector spaces over a field:A \otimes_R B := F (A \times B) / Gwhere now
F(A x B)
is the free -module generated by the cartesian product and is the -module generated by the same relations as above.
More generally, the tensor product can be defined even if the ring is non-commutative. In this case has to be a right--module and is a left--module, and instead of the last two relations above, the relation(ar,b)\sim (a,rb)is imposed. If is non-commutative, this is no longer an -module, but just an abelian group.
The universal property also carries over, slightly modified: the map
\varphi:A x B\toA ⊗ RB
defined by
(a,b)\mapstoa ⊗ b
is a middle linear map (referred to as "the canonical middle linear map".[8]); that is, it satisfies:
\begin \phi(a + a', b) &= \phi(a, b) + \phi(a', b) \\ \phi(a, b + b') &= \phi(a, b) + \phi(a, b') \\ \phi(ar, b) &= \phi(a, rb)\end
A x B.
For any middle linear map
\psi
A x B,
a unique group homomorphism of
A ⊗ RB
satisfies
\psi=f\circ\varphi,
and this property determines
\phi
within group isomorphism. See the main article for details.
Tensor product of modules over a non-commutative ring
Let A be a right R-module and B be a left R-module. Then the tensor product of A and B is an abelian group defined byA \otimes_R B := F (A \times B) / Gwhere
is a free abelian group over
A x B
and G is the subgroup of
generated by relations\begin &\forall a, a_1, a_2 \in A, \forall b, b_1, b_2 \in B, \text r \in R:\\ &(a_1,b) + (a_2,b) - (a_1 + a_2,b),\\ &(a,b_1) + (a,b_2) - (a,b_1+b_2),\\ &(ar,b) - (a,rb).\\\end
The universal property can be stated as follows. Let G be an abelian group with a map
q:A x B\toG
that is bilinear, in the sense that
\begin q(a_1 + a_2, b) &= q(a_1, b) + q(a_2, b),\\ q(a, b_1 + b_2) &= q(a, b_1) + q(a, b_2),\\ q(ar, b) &= q(a, rb).\end
Then there is a unique map
\overline{q}:A ⊗ B\toG
\overline{q}(a ⊗ b)=q(a,b)
a\inA
b\inB.
Furthermore, we can give
a module structure under some extra conditions:
If A is a (S,R)-bimodule, then
is a left S-module where
s(a ⊗ b):=(sa) ⊗ b.
If B is a (R,S)-bimodule, then
is a right S-module where
(a ⊗ b)s:=a ⊗ (bs).
If A is a (S,R)-bimodule and B is a (R,T)-bimodule, then
is a (S,T)-bimodule, where the left and right actions are defined in the same way as the previous two examples.
If R is a commutative ring, then A and B are (R,R)-bimodules where
ra:=ar
br:=rb.
By 3), we can conclude
is a (R,R)-bimodule.
Computing the tensor product
For vector spaces, the tensor product
is quickly computed since bases of of immediately determine a basis of
as was mentioned above. For modules over a general (commutative) ring, not every module is free. For example, is not a free abelian group (-module). The tensor product with is given by
M \otimes_\mathbf \mathbf/n\mathbf = M/nM.
More generally, given a presentation of some -module, that is, a number of generators
mi\inM,i\inI
together with relations
\sum_ a_ m_i = 0,\qquad a_ \in R,
the tensor product can be computed as the following cokernel:
M \otimes_R N = \operatorname \left(N^J \to N^I\right)
NJ= ⊕ jN,
and the map
NJ\toNI
is determined by sending some
n\inN
in the th copy of
aijn
(in
). Colloquially, this may be rephrased by saying that a presentation of gives rise to a presentation of
M ⊕ RN.
This is referred to by saying that the tensor product is a right exact functor. It is not in general left exact, that is, given an injective map of -modules
M1\toM2,
M_1 \otimes_R N \to M_2 \otimes_R N
is not usually injective. For example, tensoring the (injective) map given by multiplication with, with yields the zero map, which is not injective. Higher Tor functors measure the defect of the tensor product being not left exact. All higher Tor functors are assembled in the derived tensor product.
Tensor product of algebras
See main article: Tensor product of algebras. Let be a commutative ring. The tensor product of -modules applies, in particular, if and are -algebras. In this case, the tensor product
is an -algebra itself by putting(a_1 \otimes b_1) \cdot (a_2 \otimes b_2) = (a_1 \cdot a_2) \otimes (b_1 \cdot b_2).For example,R[x] \otimes_R R[y] \cong R[x, y].
A particular example is when and are fields containing a common subfield . The tensor product of fields is closely related to Galois theory: if, say,, where is some irreducible polynomial with coefficients in, the tensor product can be calculated asA \otimes_R B \cong B[x] / f(x)where now is interpreted as the same polynomial, but with its coefficients regarded as elements of . In the larger field, the polynomial may become reducible, which brings in Galois theory. For example, if is a Galois extension of, thenA \otimes_R A \cong A[x] / f(x)is isomorphic (as an -algebra) to the
A\operatorname{deg(f)}.
Eigenconfigurations of tensors
with entries in a field
represent linear maps of vector spaces, say
Kn\toKn,
and thus linear maps
\psi:Pn-1\toPn-1
of projective spaces over
is nonsingular then
is well-defined everywhere, and the eigenvectors of
correspond to the fixed points of
\psi.
The eigenconfiguration of
consists of
points in
Pn-1,
is generic and
is algebraically closed. The fixed points of nonlinear maps are the eigenvectors of tensors. Let
(a
i1i2 … id
-dimensional tensor of format
n x n x … x n
with entries
lying in an algebraically closed field
of characteristic zero. Such a tensor
A\in(Kn) ⊗
defines polynomial maps
Kn\toKn
Pn-1\toPn-1
with coordinates \psi_i(x_1, \ldots, x_n) = \sum_^n \sum_^n \cdots \sum_^n a_ x_ x_\cdots x_ \;\; \mbox i = 1, \ldots, n
Thus each of the
coordinates of
is a homogeneous polynomial
\psii
of degree
x=\left(x1,\ldots,xn\right).
The eigenvectors of
are the solutions of the constraint
\mbox \begin x_1 & x_2 & \cdots & x_n \\ \psi_1(\mathbf) & \psi_2(\mathbf) & \cdots & \psi_n(\mathbf) \end \leq 1
and the eigenconfiguration is given by the variety of the
minors of this matrix.[9]
Other examples of tensor products
Tensor product of Hilbert spaces
See main article: Tensor product of Hilbert spaces.
Hilbert spaces generalize finite-dimensional vector spaces to countably-infinite dimensions. The tensor product is still defined; it is the tensor product of Hilbert spaces.
Topological tensor product
See main article: Topological tensor product.
When the basis for a vector space is no longer countable, then the appropriate axiomatic formalization for the vector space is that of a topological vector space. The tensor product is still defined, it is the topological tensor product.
Tensor product of graded vector spaces
Some vector spaces can be decomposed into direct sums of subspaces. In such cases, the tensor product of two spaces can be decomposed into sums of products of the subspaces (in analogy to the way that multiplication distributes over addition).
Tensor product of representations
See main article: Tensor product of representations.
Vector spaces endowed with an additional multiplicative structure are called algebras. The tensor product of such algebras is described by the Littlewood–Richardson rule.
Tensor product of quadratic forms
See main article: Tensor product of quadratic forms.
Tensor product of multilinear forms
Given two multilinear forms
f(x1,...,xk)
g(x1,...,xm)
on a vector space
over the field
their tensor product is the multilinear form
(f \otimes g) (x_1,\dots,x_) = f(x_1,\dots,x_k) g(x_,\dots,x_).[10]
This is a special case of the product of tensors if they are seen as multilinear maps (see also tensors as multilinear maps). Thus the components of the tensor product of multilinear forms can be computed by the Kronecker product.
Tensor product of sheaves of modules
See main article: Sheaf of modules.
Tensor product of line bundles
See also: tensor product bundle.
Tensor product of fields
See main article: Tensor product of fields.
Tensor product of graphs
See main article: Tensor product of graphs. It should be mentioned that, though called "tensor product", this is not a tensor product of graphs in the above sense; actually it is the category-theoretic product in the category of graphs and graph homomorphisms. However it is actually the Kronecker tensor product of the adjacency matrices of the graphs. Compare also the section Tensor product of linear maps above.
Monoidal categories
The most general setting for the tensor product is the monoidal category. It captures the algebraic essence of tensoring, without making any specific reference to what is being tensored. Thus, all tensor products can be expressed as an application of the monoidal category to some particular setting, acting on some particular objects.
Quotient algebras
A number of important subspaces of the tensor algebra can be constructed as quotients: these include the exterior algebra, the symmetric algebra, the Clifford algebra, the Weyl algebra, and the universal enveloping algebra in general.
The exterior algebra is constructed from the exterior product. Given a vector space, the exterior product
V\wedgeV
is defined asV \wedge V := V \otimes V/\.Note that when the underlying field of does not have characteristic 2, then this definition is equivalent toV \wedge V := V \otimes V / \.The image of
v1 ⊗ v2
in the exterior product is usually denoted
v1\wedgev2
and satisfies, by construction,
v1\wedgev2=-v2\wedgev1.
Similar constructions are possible for
V ⊗ ... ⊗ V
(factors), giving rise to
ΛnV,
the th exterior power of . The latter notion is the basis of differential -forms.
The symmetric algebra is constructed in a similar manner, from the symmetric productV \odot V := V \otimes V / \.More generally\operatorname^n V := \underbrace_n / (\dots \otimes v_i \otimes v_ \otimes \dots - \dots \otimes v_ \otimes v_ \otimes \dots)That is, in the symmetric algebra two adjacent vectors (and therefore all of them) can be interchanged. The resulting objects are called symmetric tensors.
Tensor product in programming
Array programming languages
Array programming languages may have this pattern built in. For example, in APL the tensor product is expressed as ○.× (for example A ○.× B or A ○.× B ○.× C). In J the tensor product is the dyadic form of */ (for example a */ b or a */ b */ c).
Note that J's treatment also allows the representation of some tensor fields, as a and b may be functions instead of constants. This product of two functions is a derived function, and if a and b are differentiable, then a */ b is differentiable.
However, these kinds of notation are not universally present in array languages. Other array languages may require explicit treatment of indices (for example, MATLAB), and/or may not support higher-order functions such as the Jacobian derivative (for example, Fortran/APL).
Book: Bourbaki, Nicolas. Nicolas Bourbaki . Elements of mathematics, Algebra I. Springer-Verlag . 1989. 3-540-64243-9.
Web site: Gowers. Timothy. Tim Gowers. How to lose your fear of tensor products. dead. https://web.archive.org/web/20210507014709/https://www.dpmms.cam.ac.uk/~wtg10/tensors3.html. 23 July 2021.
Book: Grillet, Pierre A.. Abstract Algebra. 2007. Springer Science+Business Media, LLC. 978-0387715674.
Book: Halmos, Paul. Paul Halmos. Finite dimensional vector spaces. 1974 . Springer . 0-387-90093-4.
Book: Hungerford, Thomas W.. Thomas W. Hungerford. Algebra . 2003 . Springer . 0387905189.
Book: S.. Mac Lane. Saunders Mac Lane. Garrett Birkhoff . Birkhoff . G. . Algebra. AMS Chelsea. 1999. 0-8218-1646-2.
Book: M.. Aguiar. S.. Mahajan. Monoidal functors, species and Hopf algebras. CRM Monograph Series Vol 29 . 2010. 978-0-8218-4776-3.
Web site: Bibliography on the nonabelian tensor product of groups .
See Tensor or Tensor (intrinsic definition).
This similar to how the engineering use of "
(\bmodn)
" specifically returns the remainder, one of the many elements of the
equivalence class.
Book: Hazewinkel. Michiel. Gubareni. Nadezhda Mikhaĭlovna. Gubareni. Nadiya. Kirichenko. Vladimir V.. Algebras, rings and modules. 100. Springer. 2004. 978-1-4020-2690-4.
Web site: Archived copy . 2017-09-02 . live . https://web.archive.org/web/20170902184851/https://ncatlab.org/nlab/show/tensor+product . 2017-09-02 .
Analogous formulas also hold for contravariant tensors, as well as tensors of mixed variance. Although in many cases such as when there is an inner product defined, the distinction is irrelevant.
Web site: The Coevaluation on Vector Spaces. 2008-11-13. The Unapologetic Mathematician. 2017-01-26. live . https://web.archive.org/web/20170202080439/https://unapologetic.wordpress.com/2008/11/13/the-coevaluation-on-vector-spaces/. 2017-02-02.
See Compact closed category.
Book: Hungerford, Thomas W.. Algebra. Springer. 1974. 0-387-90518-9.
Abo . H. . Seigal . A. . Sturmfels . B. . Eigenconfigurations of Tensors . 2015 . math.AG . 1505.05729 .
Book: Tu, L. W. . An Introduction to Manifolds . Universitext . Springer . 25 . 978-1-4419-7399-3 . 2010.
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Tensor product".


Datacards are the portable, mass data storage devices of the Combine. They are used to record scientific and engineering data of technology and equipment designs. They are a physical object in the game that can be picked up and carried from place to place just like a blaster or a lightsaber. Only one technology or design may be recorded onto a single datacard. Recording new information onto a datacard erases the data previously held. The information on one datacard can be copied to another datacard if a character has a datapad.
The primary use of the datacard is in manufacturing. To build anything in the Combine, a datacard containing the design of the object to be built must be present at the appropriate shipyard or factory along with the required raw materials. Every ship, vehicle, droid, piece of equipment, and facility will have a datacard that defines the specifications, blueprints, and processes required to manufacture them. If the datacard is removed before construction is complete, manufacturing stops until the datacard (or an exact copy) is returned.
Generic - These datacards are available for purchase by general production factions based on faction size. They can be sold (for no return) at any time, and a new generic datacard may be purchased to replace them.
Purchasing a generic requires slots*5,000,000 credits and the total number of active members in your faction must exceed the total number of slots of all generics that your faction owns. Generics can be sold to free up slot room, in this case no money is returned.
Datacards will be central to the R&D process. To generate new, more powerful technologies (or modules), a datacard containing the existing technology will need to be present to use as a starting point. Every weapon, propulsion system, and piece of electronic equipment will have a datacard with the technical data describing the technology. To generate new equipment designs, datacards containing all the different technologies to be used in the new design will need to be present.
Datacards can have custom images purchased for them so they produce the custom image instead of the default. Once set, you have the option of turning the image on and off for production via the production menu. There is a one time fee to set a custom datacard image of 5% of the Raw Material Price (RMP) of the entity on the datacard, with a minimum of 2,000credits. Both regular and large images can be set, and all images are reviewed by the art team before approval. The form to set an image is located under the "Assign Datacard" form.
Given their essential role in manufacturing, characters will guard their datacards closely. This combination of high cost, high demand, and low availability will make them very expensive.
Datacards bring industrial espionage to the Combine in several ways. For starters, stealing a rival's datacard can keep them from manufacturing goods and developing new designs. Plus, the new owner of the datacard can use it for their manufacturing and design projects. And most devastating of all, remember that the RESEARCH variable is part of each datacard. This means a stolen datacard could have precious research and design information that makes it easier for the new owner to create advanced technology!


text:
Ice Cube says Lakers fans should give Dwight Howard 'a clean slate'
By Dane DelgadoAug 30, 2019, 10:00 AM EDT
Photo by Scott Gries/ImageDirect
Ice Cube has been a Los Angeles Lakers fan his entire life, and so he's well aware of the back-and-forth drama that occurred between the organization and Dwight Howard.
The former (and now current) Lakers big man is back with Los Angeles, and everyone is hoping it goes much better than the lone season Howard spent with the team back in 2012-13.
That includes Cube, who says that Lakers fans should give Howard "a clean slate" as he tries to help LeBron James and Anthony Davis get deep into the Western Conference finals this season.
As for what happened with the "Now This Is Going To Be Fun" Lakers? That's all in the past, according to Cube.
Via LA Times:
"I think Los Angeles should give Dwight a clean slate and let his game speak for itself," he said. "If he can help, it'll show and if he can't, it'll show. I would love for him to concentrate on what he does best — rebound, defense, blocked shots and lob dunks. If he can do that, Dwight will be all right. I'm not even going to think about the first go around. That's in the past."
If I had to guess, Lakers fans will start out a loving Howard as the season begins. They sort of don't have any other choice given the lack of depth in the frontcourt for Los Angeles.
Howard is supposed to be coming back in better physical condition, which is great given how limited he's been in staying on the floor thanks to various physical ailments as he's gotten on in age. If he can stay on the court, Howard's numbers will likely prove pretty useful. If not, it will be like everything else in Lakerland, and he could quickly become a scapegoat.
The Lakers are not going to be a contender this year. Anyone telling you otherwise is getting their hopes up, and apparently didn't watch last season. LeBron is still great, but the likelihood for injury between he and Davis is extremely high. With the star power in the rest of the Western Conference, the Lakers just come up short.
In any case, and as always, Los Angeles is going to be a reality show of a team and it will be entertaining to watch no matter what. Hopefully for Howard's sake, he'll be able to come in, contribute, and duck any serious criticism. That might be the best-case scenario for the former All-Star.


text:
Q: Structr drag-n-drop shared component fails structr (2.0.0 and 2.0.1) - drag-n-drop component from project to shared component fails. Ubuntu 14.04, Chrome Version 54.0.2840.100 (64-bit), java version "1.8.0_20".
Selects in the Pages Tree View, visible dragging across the page but then disappears with release of the mouse button. Attempting to work the demo project, moving the head element to shared components.
Other components and components from the Welcome page fail as well.
Just in case, three warnings from the server log:
17:08:18.051 [main] WARN  o.s.n.StructrWrappingNeoServerBootstrapper - The specified file for server configuration [/etc/neo] does not exist. Using the default non-user-defined server configuration.
17:08:18.057 [main] WARN  o.s.n.StructrWrappingNeoServerBootstrapper - No database tuning file explicitly set, defaulting to [/etc/neo4j.properties]
17:08:18.057 [main] WARN  o.s.n.StructrWrappingNeoServerBootstrapper - The specified file for database performance tuning properties [/etc/neo4j.properties] does not exist.

Suggestions?

A: Missed a subtlety in the documentation. Selecting the element means pointing to any non-icon part of the element bar, then drag-n-drop. (This was after a careful re-install of the latest Java.)
See Step 16 of https://support.structr.com/article/232, which doesn't mention the significance of the mouse pointer location.

redpajama_set_name: RedPajamaStackExchange


Located near Florida's southern tip and southeast of Naples, Everglades City is known as the gateway to Ten Thousand Islands. Canoe the mangrove tunnels. Take a fishing and backcountry excursion. Hike through the Big Cypress National Preserve near Everglades City. And try not to miss the Everglades Seafood Festival, a taste of Florida as it used to be. Generally held in February, the festival is as laid-back as a day on the river, yet as lively as a tune on the banjo. It's as real as gators in the Everglades City swamp. It's a place to snag a seat on a carnival ride, relax with the family and raise a mug of suds to the stars – and, of course, sample sumptuous Everglades City seafood, fresh and firm and perfect.
Best access to old Florida swamp life.


We take the security of our customers seriously and try to resolve any security issues as soon as they are reported.
If you found any security issues with CloudLinux software or its components, including the Imunify360 and CloudLinux Backup, we appreciate your prompt disclosure of such issues.
Imunify360 software, including its components, such as Firewall, IDS/IPS, Malware scanning, etc.


BARCELONA have stepped up their interest in highly-rated Uruguayan youngster Juan Manuel Sanabria, according to reports.
Barca scout Pep Boada is said to have been tracking the 16-year-old midfielder for some time and the La Liga Champions are now set for talks with Uruguayan club Nacional.
Spanish newspaper Mundo Deportivo claim Barca will begin negotiations with Nacional's representatives this week in the hope of luring the player to the Nou Camp.
But it's not just in the Premier League that Sanabria is causing a stir, as La Liga giants Atletico Madrid and Villarreal are also said to be hot on his heels.
Mundo Deportivo say he favours a move to Barca where he would team-up with fellow Uruguayan and former Nacional star Luis Suarez.
However, he is said to be mature for his age and is seeking a club that will give him the best platform to develop, rather than the highest-paying contract.
Sanabria is the captain of Uruguay's U17s and led the side in last year's U17 World Cup in Chile.


How Can I Be Connected?
Grand Rapids Child Discover Center welcomes you to our community and encourages your participation in the life of our school. Your involvement fulfills our mission to continuously expand the potential of children, the experiences of educators and the involvement of parents within a diverse community of Grand Rapids, Michigan.


text:
Chinese cities are competing to woo overseas entrepreneurs
The coffee's on us
WHEN Maria Veikhman, founder of SCORISTA, a Russian credit-scoring startup, was considering expansion abroad, China immediately came to mind. She believes the scope there is vast, for two-fifths of Chinese have no credit records. Ms Veikhman settled in Tianfu Software Park, a state-owned incubator in Chengdu, capital of Sichuan province where city authorities "offer almost everything for free". Complementary facilities range from office space, basic furniture and logistics services to detailed guidance on entrepreneurial methods.
Chengdu aims to catch up with Beijing, Shanghai, and Shenzhen, which at present are in a different entrepreneurial league—together they have over a hundred unicorns, or private startups worth over $1bn. The south-western city allocated 200m yuan ($30m) in 2016 to an innovation-and-startup fund for overseas founders, and hands out up to 1m yuan in cash to well-capitalised foreign startups and joint ventures. If the...Continue reading
Two in one: human placenta stem cells hold a dual benefit
Europe's flourishing gunmakers
Feb 22 Approved — Business — Business — Business and finance


In the result of investigatory actions of great volume conducted within the criminal case investigated in Lori Regional Investigative Department of the RA Investigative Committee circumstances of murder of the resident of Yerevan Arsen Sardaryan, who had been missing, were found out. 30 year-old resident of Alaverdi town was arrested on suspicion of committing murder.
On April 5, 2019 the resident of Yerevan submitted a report to Police Division of Malatia informing that on April 4, 2019 at about 16:30 his brother Arsen Sardaryan, born in 1973, drove to Spitak town in his car of «Mercedes» make and had been missing since then.
In the result of operative-intelligence, as well as operative-searching measures of great volume the mentioned car of «Mercedes» make was found. It was found out that the person driving the car at the moment, who was a resident of Alaverdi town, born in 1989, on April 4, 2019 was in the car with him leaving for Alaverdi.
Data were obtained that on the way at about 20:30 30 year-old resident of Alaverdi argued with Arsen Sardaryan during which he stabbed the latter on the head causing bodily injuries incompatible with life. In order to conceal the crime traces he took the man's body in the same car to an empty area in the administrative district of Alaverdi community and burnt. Then he took the burnt body to the field next to the road to Akori village of Lori and buried it under soil and leaves.
On April 7, 2019 a criminal case was initiated in Lori Regional Investigative Department of the RA Investigative Committee according to the Part 1 of the Article 104 of RA Criminal Code.
On the same day 30 year-old resident of Alaverdi was arrested and gave testimony pointing out the place where he had buried the body.
Forensic medical examination of the body, forensic trace and forensic biological examinations were commissioned to find out circumstances significant for the criminal case.


text:
https://www.thehour.com/norwalk/article/Weather-can-t-keep-River-Ramble-down-8186278.php
Weather can't keep River Ramble down
By MATT COYNE
Hour Staff Writer
Published 1:45 pm EDT, Saturday, July 13, 2013
Photo: (C)2013, The Hour Newspapers, all rights reserved
ROWAYTON -- Cloudy skies and some drizzles couldn't keep residents away, as the Rowayton Civic Association held its annual River Ramble.
The 43rd annual event saw food and drink tents, vendor tables and children's games pop up at Pinkney Park along Five Mile River Saturday morning.
"I can't even tell you how many residents tell me during the year that they'll never miss a River Ramble, ever. And that's adults without kids. That's their highlight and they never miss it," said Wendy Ford, the Civic Association's president. "We've got a raw bar, we've got beer, we've got food, we've got music. We're right on the water. What more could you ask for?"
The weather held off for most of the day, except a brief drizzle in the afternoon, before the live auction, one of the Ramble's most popular attractions.
The weather, however, did not hold up during set-up.
"It's incredible, from people showing up last night in the rain to people this morning at 7 a.m. showing up in the rain. The community just totally rallies around it," said event co-chairman Nora King. "It really, honestly takes a village. I've been totally humbled by how many people pulled together to make this happen."
King, who joined the Civic Association in January after serving on the Common Council, said putting on the Ramble was more work than putting on her wedding.
"We used to come every year, I have no idea what it entailed though. I used to just bring my kids and have a good time," she said. "(This morning) I was up at 3 a.m. watching the Weather Channel, because I couldn't sleep."
The Ramble is one of the Civic Association's three big fundraisers, along with Comedy Night and the biennial house tour, which focuses on one neighborhood within Rowayton. The money raised goes toward the Civic Association's myriad of community events, like the Memorial Day Parade, fireworks, Summer Music Festival, soccer and swim races and grants for local community groups and schools.
"We've been around a long time. Our mission is to improve the lives of our community," said Ford. "We are able to give back to the community by having these events."


text:
County Council Opposes BPP Tax Elimination
David Slone, Staff Writer
Like the Warsaw Common Council earlier this week, the Kosciusko County Council on Thursday approved a resolution opposing legislation regarding the business personal property tax elimination.
Council President Sue Ann Mitchell provided the resolution and said, "Personal property elimination at the state legislature has become a really big topic. AIM - the unit that represents municipalities - and the (Association of Indiana Counties) have worked together to try to provide information and to provide the data in order to help the legislators understand what the impact would be of reducing or eliminating personal property."
She said the AIC has discussed that it would be beneficial for those who were interested to sign a resolution explaining what the impact would be.
"There is no way to break out what they're talking about right now, is eliminating the 30% floor on personal property, which means when people got done depreciating their personal property, it would fall off the records and there would be no further taxes collected on those items," Mitchell explained.
"Sounds like a great idea. We don't oppose them doing it. What we oppose is not making the money available back to local government that their legislation is taking away."
The slogan has become "Replace, Don't Just Reduce."
Mitchell said County Auditor Michelle Puckett worked on the figures Thursday and "if we lost all personal property - this is all, not just that 30% floor; but you also have to remember that every year they have decreased the dollar threshold that you have to file in order to pay your taxes. So that reducing threshold - currently, in Kosciusko County, we collect over $60 million in personal property - so if that funding mechanism went away, and because of the tax caps, the shift would be from those businesses that had those equipments over to homeowners. That's a pretty big ask for homeowners."
Council Vice President Joni Truex said, "I don't think our legislators at the state have really considered the fact that businesses can raise the price on products in most cases in order to offset some of their expenses, like personal property taxes, etc. Homeowners have no way of raising money from their home. It is what it is, and, eventually, we could price out, just based on the taxes of the house, to where people couldn't even afford to own a house."
Councilman Ernie Wiggins said, "I just think it's important that we contact our legislators. Warsaw City Council already approved their legislation, and I think about every cities and towns and counties should be supportive. This is just a bad bill. Period."
Councilwoman Kimberly Cates said it may have been a great idea to the state legislators, but "I don't think they realized the dollar amount that it's actually going to affect the counties and county income eventually."
"In all due respect," Wiggins said, "They should have done a fiscal impact study before the introduced the bill. So I think somebody just wanted to push this agenda."
Mitchell said AIC spent $70,000 to do a study in order to provide the overall information. "That tells you how important the board of AIC thought it was. And AIM has provided a lot of the documentation, the public relation stuff, to get that information out."
Cates made a motion to pass the resolution, Wiggins seconded it and it was approved 7-0.
Mitchell also brought up another item that wasn't on the Council's agenda.
"One of the things that we need to talk about, be prepared for, what we're going to do at next month's meeting, is where we're at with the agreement for the (Kosciusko County Convention Recreation Visitors Commission), and how we're going to handle the change in what the duties are that the auditor is going to do for them," she said. "Some of you may realize that things had changed with the auditor's office and their interaction."
She said the Council needed to decide if they wanted to attach "something to it financially" in exchange for the auditor's office doing the work, providing the supplies and advertising "and all that."
Puckett said her office helps out many organizations as far as secretarial skills and being comptroller for them. That includes the Kosciusko County Solid Waste Management District, Redevelopment Commission, County Council and Commissioners and the KCCRVC.
"Over the years, our duties for the Commission have changed, and where we're at right now is that we have a need to perform a few more services than what we were previously," she said.
Puckett said they have an interlocal agreement with the KCCRVC that "is more vague and less defined."
"And in recent conversations with some things that have happened, I am more than willing to step forward and provide more assistance to them in the financial side, as well as in the secretarial side, but felt it was very important to have those details outlined in a new interlocal agreement,"?she stated.
Historically, the auditor's office did not collect a fee from the KCCRVC for the service it provided, she said, but the office does pay to advertise the meetings and use the county's paper, ink, printers, etc. Puckett said $500 a year would be fair to compensate the county for the supplies.
"But with the interlocal agreement organized the way it is, I don't feel there would be much more of a hindrance on the workload of my office and we are committed to fill that in our regular work day with my current staff," she said.
Councilman Mike Long asked if it would be an annual agreement and Puckett said yes. The KCCRVC, County Commissioners and Council all have to approve it.
The KCCRVC meets six times a year and their next meeting is in March, before the Council's next meeting, Mitchell said.
Councilman Jon Garber made a motion to approve the $500 fee and the Council unanimously approved it. The Commissioners also will need to approve it.
In other business, the Council approved:
• An additional appropriation of a $40,000 Emergency Management Performance grant from 2021, as requested by Kosciusko County Emergency Management Director Ed Rock. He said the county will use the money to enhance its cybersecurity.
• An Indiana Department of Natural Resources grant proposal for $15,000, as requested by Kosciusko County Sheriff's Office Chief Deputy Shane Bucher. The money, if awarded, will go toward the KCSO Marine Patrol Division so they can patrol Lake Wawasee, Syracuse Lake and Tippecanoe Lake and their chains. The County Commissioners previously approved the grant application.
• An additional appropriation of $215,299.50 from the Edward Byrne Memorial Justice Assistance Grant for new KCSO radios, as requested by Bucher. He said the radios have come in to J&K Communications and starting Tuesday the KCSO cars will have the new radios put in on a rotational basis.
Mitchell said, "Just to clarify, what we had appropriated out of the (American Rescue Plan Act) money, or what we said we were going to spend, that was already reduced by this amount because by the time we did that, we knew that they had those grants."
Along with KCSO's grant, other departments in the county receiving funding from the JAG program were Pierceton, Claypool and Winona Lake police.
"So that helped reduce what we were spending out of the ARPA money," Mitchell said.
• An additional appropriation of $24,000 from the Patrick Leahy Bulletproof Vest Partnership grant for bulletproof vests; $30,000 from a Comprehensive Hoosier Highways Injury Reduction Program (CHIRP) Click It To Live It 2022 grant and $22,500 from a CHIRP DUI Task Force 2022 grant for overtime patrolling for seatbelt and DUI enforcement, as requested by Bucher.
• An additional appropriation of $12,225 for insurance vehicle replacement as requested by Bucher. He said last fall an officer in his patrol vehicle hit a deer and the insurance company ended up totaling the vehicle out for $12,225. Bucher said the KCSO wanted to move that money from the general fund into the KCSO insurance replacement to use for new vehicles.
• A transfer of $15,000 from sheriff maintenance contracts to sheriff dues and subscriptions, as requested by Bucher. He said the transfer was to compensate additional subscriptions the KCSO has and to pay the subscriptions "in the correct account."
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until every tax dollar local, state and federal is accounted for Don't raise my taxes. there is no reason in this computer age that every penny is not accounted for to the American people. This comment has been hidden due to low approval.
Neil Bammerlin


NUTLEY, NJ - A resolution was approved objecting the closing of the Highfield Lane at grade rail crossing in Nutley during the Tuesday, March 5 Board of Commissioners public meeting.
The resolution, which was unanimously passed by the Board of Commissioners, states that Nutley strongly objects to the arbitrary and unpredictable selection by Norfolk Southern Corp. to closing the Highfield Lane grade rail crossing to vehicular and pedestrian traffic. Norfolk Southern Corp. has recently petitioned the New Jersey Department of Transportation for the closure.
According to the resolution, there has been no train traffic on Norfolk Southern's Newark Industrial Track crossing Highfield Lane for several years.
The resolution states closing of Highfield Lane will negatively impact emergency response time in the area east of the existing crossing. The County of Passaic will also be constructing a new bridge at nearby Kingsland Street in Clifton which Kingsland Street will be closed for 12 to 18 months, thus further impacting traffic patterns in that area.
A letter dated Jan. 8, 2019 was sent from Prism Capital Partners, LLC objecting to the arbitrary closing of Highfield Lane grade rail crossing. The Township of Nutley also sent a letter to Norfolk Southern Corp. and New Jersey DOT on Nov. 18, 2018 objecting to the closer and withdrew their support for the conversion of private rail crossings at the former Hoffman-La Roche campus in Clifton into permanent public crossings.
A resolution was passed to acknowledge March as National Women's History Month. Nutley observed International Women's Day on March 8.
A resolution to authorize the execution of a property tax appeal settlement agreement between the Township of Nutley and PB Nutclif Master, LLC.
PB Nutclif Master, LLC has acquired the property known as the former Hoffman-La Roche site. According to Revenue and Finance Commissioner Thomas J Evans, there were outstanding tax appeals for the years 2016, 2017 and 2018 filed by Hoffman-La Roche.
Angelo Cifelli Jr. Esq. of the law firm of Piro, Zinna, Cifelli, Paris & Genitempo in Nutley was awarded a one year contract from Jan. 1, 2019 through Dec. 31, 2019 not to exceed $17,000 to provide legal advice and services in connection with the township's Land Use Policies and Litigation.
Quality Fence of Peekskill, Inc. in Peekskill, New York was awarded a contract for $29,695 to replace the fencing around the American Little League baseball field.
The Township of Nutley is submitting the 2018 Municipal Recycling Tonnage Grant Application to the New Jersey Department of Environmental Protection and designates Salvatore Ferraro and recycling coordinator. It is intent of the Mandatory Source Separation and Recycling Act to use the tonnage grants to develop new municipal recycling programs and to continue to expand existing programs.
A refund of 2019 tax charges in the amount of $2,882.36 was awarded to 204 Regents Drive.
Kevin Hussey, an employee in the Department of Public Safety was granted an extension of a medical leave of absence without pay effective Feb. 18, 019 through April 1, 2019.
Raffle applications were approved for the Nutley-Belleville Columbus Day & Italian Heritage Month Parade Committee for on-premise merchandise and 50/50 cash raffles for Sunday, March 31 and an off-premise 50/50 cash raffle for Oct. 13; and Washington School PTO for an off-premise 50/50 cash raffle and on premise merchandise and 50/50 cash raffle for April 27.


Mediacorp has the widest range of media platforms in Singapore spanning digital, television, radio, print and out-of-homemedia. Its mission is to engage, entertain and enrich audiences by harnessing the power of creativity.
Mediacorp pioneered the development of Singapore's broadcasting industry, with the radio broadcast in 1936 and television broadcast in 1963.
For digital, it is the 4th largest web property in Singapore with 65% market reach. Operating 14 premium brands across News & Information, Entertainment, and Lifestyle.