Maria was born and raised in Colombia, South America. She moved to the USA in 2000. In 2002, Maria moved to Wyoming where she started her teaching career as a Spanish teacher assistant. In 2013, she graduated from Montana State University with a Masters degree in Education. Maria is looking forward to using her skills as a native Spanish speaker to share the beauty of the Hispanic Culture with our HPA students. Her favorite hobbies include listening and dancing to Latin music such as Salsa, Merengue, Bachata.
Jason Gavaldón is the art teacher at High Point Academy. Jason earned his bachelor's degree in finance from Pace University in New York City and a master of arts in music from the Brooklyn College Conservatory. He taught for Free Arts New York, a program that provides art and expression to inner city kids who live in impoverished areas, from 2013-2014. Jason has performed in New York City as a dj and musician. He moved to Denver to produce music, explore the mountains, and to begin his PhD studies.
Jessica Lawdan is thrilled to be joining the staff as HPA's first music teacher! She is originally from Chicago, IL and graduated from St. Olaf College with a Bachelor's of Music in Vocal Music Education. She has been singing in choirs, a cappella groups, caroling groups, and on her own for as long as she can remember and can usually be found doing any given activity while singing, rapping, beatboxing, or making some kind of noise (just ask anyone who knows her). She is an active musician around the Denver-metro area, and enjoys singing and playing her original music at a variety of open mics around town. Ms. Lawdan is also the Music Director for the Vintage Youth Theater's summer musical theatre camp in Aurora and loves teaching kids to act and express themselves on stage. She has lived in Colorado since 2015 and hopes to spend many more years here exploring the beautiful scenery, meeting great people, and impacting students to become their selves through music.
Dane Tuck is the physical education teacher at High Point Academy. Mr. Tuck was born and raised in London, England and is an all around sports fan since he could crawl. Crystal Palace is his #1 soccer team in England. Go Eagles! Mr. Tuck graduated from St. Mary's University in physical education and sport with education and social sciences in 2011. Since graduating he has been coaching sports and teaching physical education, using a variety of different sports played worldwide. Mr. Tuck has lived in Denver, CO for one year and enjoys exploring the mountains, cheering for the Broncos and playing rugby in his spare time.
Marc Vigil is the physical education teacher at High Point Academy. Mr. Vigil was born and raised in Leadville, Colorado. Coaching right out of high school, Mr. Vigil knew he wanted to coach and work with kids and sports and realized teaching and coaching would be a perfect fit. After attending the University of Northern Colorado, he graduated with a Bachelor of Science in Sport and Exercise Science in 2008. He brings with him multiple years of teaching and coaching experience and is ready to bring new and innovative ideas to the PE curriculum. Mr. Vigil is a firm believer in trying your hardest and doing your best, controlling the controllable, and always being a team player!


text:
Profile: Hand Spring Puppet Company
By Daisy on	 June 21, 2002 Artist info - stage
Founded in 1981 by four graduates of the Michaelis School of Fine Art in Cape Town, this group is still run by two of the co-founders, Adrian Kohler and Basil Jones.
Originally set up to create shows for children, Handspring graduated to adult audiences and has since become regarded as one of the greatest puppetry companies in the world, working with many South African directors including Malcolm Purkeyand Barney Simon as well as artist William Kentridge.
Handspring has performed at many international festivals including Edinburgh, the Avignon Festival, the Next Wave Festival at BAM in New York, The African Odyssey Festival at the Kennedy Centre in Washington, Theatre d' Automne in Paris, Theatre der Welt in Germany, as well as in Hong Kong, Singapore, Adelaide, Zurich and Bogota.
With Thys Stander as the company's chief puppet maker Handspring has won an Olivier Award, an Evening Standard Theatre Award and London Critics' Circle Theatre Award. For the production of War Horse, the company has received an Outer Critics Circle Special Achievement Award for Puppet Design, Fabrication and Direction and they are also a recipient of the 2011 Special Tony Award.
www.handspringpuppet.co.za
Review: War Horse
Review: Behind The Scenes Of Handspring
Profile: Lara Foot
Profile: John Vlismas
Previous ArticleProfile: Marc Lottering
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redpajama_set_name: RedPajamaCommonCrawl


text: Q: How coordinate system shifting is related to similarity transformations? I know that coordinate system shifting can be represented using matrices. But how exactly are similarity transformations related to coordinate shifts ?

redpajama_set_name: RedPajamaStackExchange


American football is a unique sport in both the brevity of its season and the preponderance of injuries due to its extremely violent nature. Both of these feature make prognostication very difficult, because 16 games are erratic enough even if you know very well that the best players will compete.
Instead of trying to project win totals for individual teams, I have used numberFire's projections to set the number of division and wild card wins each team will likely need to qualify the playoffs. These targets are adjustable via the parameters on the right (if you don't see the parameters, maximize your browser window).
Some are having problems interacting with the below graph. If this is the case for you, please click here to view the visualization on its own, where you can use the filters and see the hover-over text.
Given all the randomness created by injuries and a short season, I find it most helpful to simply track what NFL teams need to accomplish over their remaining games rather than trying to quantify odds for a few games whose key players may not even be fit to perform.
If you like this visualization, please bookmark, as I will aim to update it regularly.


text:
Fab, No Fluke! Why Fans Should Believe Cleveland Indians Hot Start
April Showers Bring May Fireworks
By Eddy Jansen May 2, 2011, 10:24am CDT
Share All sharing options for: Fab, No Fluke! Why Fans Should Believe Cleveland Indians Hot Start
Cleveland sports teams have been known to portray their city's weather: One day you get something, the next you get something completely different. It does add a spicy sense of excitement, in a sense that fans never what they're going to get. But it finally looks like the Indians are going to add one crucial element to their performance: consistency. That is one of many reasons to believe that the Tribe is for real, and not just a coincidence.
1. They are winning close games
Good teams just get it done. No matter what way, shape, or form, whether impressive or ugly, walk off of not, the Indians have gotten the end result to be the same. The seventh, eighth, ninth and every inning after that has been an area where the Indians excel with a high level of comfortability and tranquility. They've gone a respectable 7-5 in games that were decided by three runs or less.
2. They are winning the low scoring games
There have been an abundance of pitchers duels this season, and the Indians bats have given just enough runs of support. They are 7-6 in games where they have posted four runs or fewer. In previous years, if the Tribe didn't score higher than that mark: forget it. They stood no chance; this season is different. Their pitching, starting and relieving, hasn't been bad either. Their team ERA is ninth in the big leagues.
3. The lineup in coming together without the presence of a big bat
In manager Manny Acta 's everyday lineup card, there really isn't a go-to, clutch, experienced hitter. But that doesn't matter. Hitters 1-through-9 have done their fair share of damage to oppose pitchers. The Indians as a team are 4th in MLB in batting average, ahead of the Texas Rangers, Philadelphia Phillies, and New York Yankees.
4. Chicks dig the long ball!
Cleveland's 34 dingers are third best in the league. They have seven players with four homeruns or more in the season's opening month. Solid starts to the season's of Jack Hannahan and Grady Sizemore (after his left knee was fully recovered from surgery) have played a large part of the fireworks at Progressive Field.
5. Something special was bound to happen in a season dedicated to Bob Feller
The man who represented the Cleveland Indians franchise for the past century, nearly, had his number patched on each 2011 Indians jersey for the entire season. The team has gotten the post-Feller era off on a good note, and it should continue through the next couple of months, at least.


text:
Former BJP data analyst on how the party wins elections and influences people
Shivam Shankar Singh, a former data analyst with the BJP explains how the party used technology to fuel casteist and communal strife to win elections.
Courtesy Shivam Shankar Singh
In June 2018, Shivam Shankar Singh, a former data analyst with the Bharatiya Janata Party, wrote a post on the online blogging platform Medium, titled "Why I am resigning from BJP." In it, he said he took the decision to leave the party as "the negatives of this Narendra Modi and Amit Shah government now outweigh the positives for me."
Singh worked for the BJP ahead of the assembly elections in Manipur and Tripura, held in 2017 and 2018 respectively. He used data to help chalk out election strategies and target voter-groups through social media. During this time, he wrote, he noticed that the BJP was "spreading some specific messages with incredibly effective propaganda." He also noticed drastic differences between what the party promised and what it was delivering. Singh began to grow disillusioned with the party. He wrote in his post that the BJP was "pushing the national discourse in a dark corner."
Singh is now working with the mahagathbandhan, a grand coalition of political parties in Bihar, which opposes the BJP. Earlier this month, Tushar Dhara, a reporting fellow at The Caravan, met him in Delhi. Singh detailed how the BJP used technology to fuel casteist and communal strife to win elections. An edited excerpt from the conversation is reproduced below.
When I first campaigned for the BJP in 2013, there was a narrative about corruption in the country as the 2G, Coalgate, Commonwealth scams had just come to light. Modi presented himself as a great administrator, and at that time, that is what the country was looking for. There was a group set up by the election strategist Prashant Kishore, called Citizens for Accountable Governance. Young professionals from technology and consultancy services, such as KPMG and Ernst & Young, had also left their jobs and joined the CAG to work for Modi's campaign.
That is the background in which I joined the BJP. I worked for them informally—wrote a few articles and social media posts. The BJP's data set-up was very strong in 2014. But the structure soon collapsed because the party felt that they had won the election because of Modi. The party's opinion was that data was not needed. Then something major happened—in 2015, it lost the assembly elections in Delhi and Bihar. The prime minister went to Bihar and said at a rally, "Have you got electricity? Have you got roads?" He expected people to say no, but the crowd said, "Yes!" Even a basic survey would have told you to not raise this issue. The BJP ran blind and lost miserably. This is when they started to build their data structure again. By 2016, they became strong again.
That year, I joined Kishor's political advocacy group, the Indian Political Action Committee, and worked in Punjab for for a few months. [For the then upcoming Punjab legislative assembly election, the IPAC was campaigning for the Congress.]The perception was that the Congress's chief ministerial candidate Amarinder Singh, who was called a "Maharaja," is lazy and does not work too hard. The Congress's biggest weakness has been that their cadre does not exist in most places and where it does, it does not work because they are not emotionally driven.
Keywords: Elections 2019 BJP IT cell Bharatiya Janata Party Manipur Congress Tripura Assembly Elections


An innovative working-capital initiative at Bayer took on the challenge of increasing the days payable outstanding – while making suppliers better off at the same time.
The supplier financing programme in the German-based chemicals and pharmaceuticals group was initiated as part of a broad working capital initiative launched in 2008. This took aim att expanding the days payable outstanding (DPO), without any negative impact on the suppliers.
The story is told in a contributed case-describing article, at industry website Treasury Management, written by Michael Friede, who is the head of global procurement intelligence for Bayer's Materialscience subsidiary.
Already the pilot phase of the supplier finance programme, launched in the German part of Bayer Materialscience in late 2009, was deemed highly successful.
A core purpose of the solution is to let suppliers reap benefit from giant Bayer's good access to capital. Simultaneously, Bayer gains advantages in terms of operational stability, as the supply chain risks were mitigated. Also internal processes could be made more efficient.


Kanye West has a long list of awards — He's won Grammys, VMAs, and BET Awards. Now he can add another EPIC award to his trophy room.
The rapper-turned- fashion designer took home the 2015 Shoe of the Year award for his adidas Yeezy Boost 350 at the Footwear News 29th Annual Achievement Awards. And of course, in true Kanye fashion, he delivered an epic acceptance speech that spanned 20 minutes, thanking everyone from himself, to little Nori & his soon-coming son.
During his lengthy time spent at the podium, he joked about his headline-making rant at the MTV Video Music Awards, in which he announced his 2020 bid for president. "If you don't like 10-minute, profanity-ridden speeches that end with a presidential bid, you can go to the bathroom," he said.
His very pregnant & sick wife Kim Kardashian was not present at the award ceremony due to a sinus infection & the flu, but was at home cheering him on. "So well deserved!!!! Congrats babe on winning Shoe of the Year Award! I see how hard you work and how meaningful this award is!" she tweeted.


text:
THEATER MASTERS PRESENT "TAKE TEN"
Aspen's acclaimed Theater Masters will present its second annual "Take 10," an evening of nine 10-minute plays from aspiring playwrights across the country as part of its National MFA Playwrights competition on April 15-18th at Off Broadway's 59 East 59 Theaters (59 East 59th Street). "Actor John Lithgow, Theater Masters' Associate Director Pesha Rudnick read over 100 scripts submitted from nine of the top graduate play writing programs to find these remarkable plays from some of the finest young playwrights this country is developing" said Julia Hansen, Theater Masters founder / Artistic Director – and former President of New York's Drama League. "In many ways, how I started the Drama League's "Director's Project" is how I am now developing new playwrights and giving them their first chance to have their work receive a New York premiere."The 10-minutes plays are: Back Story by Benjamin Viccellio from Northwestern University - when life imitates art, one diva loses her sense of reality after too many years in the theater; The Big Exit by John-Paul Nickel from Carnegie Mellon - three contestants on America's next hit reality TV show are in for the surprise of their lives; The Echo, by Jennifer Fawcett from the University of Iowa - a tragic accident haunts the adult soul of a childhood survivor; Fernando and the Killer Queen by Kyle J. Schmidt from the University of Texas - Revolution, mayhem and l'esprit d'Abba ignite the final moments of the ruthless Killer Queen; Frogs by Sam Marks from Brown University - after all an night house party, two young women sit in the kitchen in the dawn light and assess the damage of the previous night, youth, and the past; The Hiding Place by Melisa Tien from Columbia University - a menacing, lyric glimpse at power, sexuality and the workplace; Higher Up by Annette Lee from UCLA - when the 'new guy' shows up for work, two businessmen experience firsthand what it's like to be in the dog house; Man up and Away by Joe Tracz from NYU - on the night of his Going Away to War party, Jordan says goodbye to his little brother Vic -- in the worst possible way; Rocky Road by Stephanie Andrea Timm from UCSD - one family's zany attempt to bring their son home from the war in Iraq.A sampling of Theatre Masters' casting and directors includes the following: Back Story, directed by Pesha Rudnick – Jacob Knoll as Man; The Big Exit, directed by Heath Cullens – Nick Sullivan as Bernie, Anna O'Donoghue as PA and Juliette Bennett as Cassie; The Echo, directed by Wesley Savick; Fernando and the Killer Queen, directed by Wesley Savick – Ivan Quintanilla as Fernando, Nick Sullivan as Narrator; Frogs, directed by Heath Cullens – Hannah Cabell as Christina, Anna O'Donoghue as Violet; The Hiding Place, directed by Wesley Savick; Higher Up, directed by Pesha Rudnick – Hannah Cabell as Fifi, Ivan Quintanilla as Buster; Man up and Away, directed by Heath Cullens – Jacob Knoll as Jordan; Rocky Road, directed by Pesha Rudnick – Anna O'Donoghue as Daughter, Birgit Huppuch as Mother. (more casting below)Based in Aspen, Colorado, Theater Masters is a non-profit organization whose core mission is to seek and nurture the next generation of artists for the American Theatre and present their work."Take 10" performance schedule is as follows: Tuesday, April 15th at 8PM, April 16th at 7:30PM, April 17th at 4 & 7:30PM, April 18th at 8PM. All tickets are $18 and can be purchased from the Box Office (box office hours: Mon. 12-6; Tues. - Sat. 12-8:30; Sun. 12-7:30), by phone through Ticket Central at 212-279-4200 or online at www.ticketcentral.com
GOSSIP JAMES EDSTROM JOHN LITHGOW JULIETTE BENNETT
I love you, crazy, zany John! Keep on acting!!!


text:
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redpajama_set_name: RedPajamaGithub


If you've ever lived in a humid area, you're probably familiar with a phenomenon known as atmospheric corrosion — a common result of which is everyday rust. This type of metal corrosion happens frequently enough that professionals in construction and manufacturing regularly use some form of corrosion protection, even if it's simply applying a coating to the metal. To efficiently analyze the corrosion process and optimize prevention techniques, engineers can use the COMSOL Multiphysics® software.
Atmospheric corrosion is an electrochemical phenomenon that happens when metal comes into contact with an electrolyte (such as water). Even a small film of moisture is enough to do a lot of structural damage over time. You may have seen this effect after a long winter — the bike you've accidentally left outside showing signs of rust when the snow starts to melt. Other factors that contribute to atmospheric corrosion include air pollution in cities and salt in marine environments.
Since environmental factors are not easy to predict or control, many engineers employ methods to protect their designs against atmospheric corrosion. For instance, depending on the environment, manufacturers may choose to use metals that are more naturally resistant to corrosion (like copper and aluminum) over others (like steel and iron). Other forms of protection include applying a protective coating to the design, such as a less noble metal (e.g., zinc-galvanized steel, tin-plated copper); paint; or anticorrosive material.
Another strategy for reducing atmospheric corrosion is to employ cathodic protection. A metal in contact with an electrolyte has cathodic areas where oxidation (protection) occurs and anodic areas where reduction (corrosion) occurs. The metal that is corroded is typically oxidized, and the electrons that are released are involved in cathodic reactions. While techniques vary, cathodic protection involves supplying electrons to the metal from an external source (like electric current). Cathodic protection is useful for common environments, such as where the metal would be exposed to water.
For example, by applying a metal coating such as in zinc-galvanized steel, the zinc will become an anode, since it is less noble, and protect the cathodic steel from corrosion whenever the steel is exposed to an electrolyte through damage in the zinc layer. There are other methods of cathodic protection, such as using sacrificial anodes or impressed current, but these methods are only reliable when the object is submerged in the electrolyte (e.g., water).
In certain environments, anodic protection can also be used. This method involves biasing the metal into a passive region by applying a controlled, small anodic current. This current will create a thin, passivating film layer that "chokes" the anodic corrosion reaction. It is commonly used in extremely corrosive environments, such as when stainless steel is exposed to phosphoric acid.
Using COMSOL Multiphysics and the add-on Corrosion Module, engineers can evaluate cathodic and anodic protection, electrolyte potential, and corrosion reactions in a design. As an example, let's take a look at a model of a busbar, a bar used to distribute power in buildings, vehicles, and more — thus subject to atmospheric corrosion in a number of different environments.
The busbar geometry, consisting of a copper flange, aluminum flange, and zinc nut and bolt.
In this example, the latter reaction (the reduction) is restricted by the oxygen transport through the film, with the limiting current density being dependent on the film thickness, oxygen solubility, and oxygen diffusivity.
Once the geometry is set up, the busbar is exposed to humidified air to begin the corrosion process. First, you can use the Secondary Current Distribution interface to solve for the electric potential in the electrode domain. Then, you can use the Current Distribution, Shell interface to solve for the electrolyte potential in the thin electrode layer.
The thickness of the electrolyte film depends on both the salt load density and relative humidity, while the conductivity and oxygen solubility depend on the relative humidity of the surrounding air. To account for the dependence on the relative humidity, you can take the same expressions used in the general Atmospheric Corrosion model to obtain the electrolyte film thickness, electrolyte conductivity, and oxygen solubility. The oxygen diffusivity is assumed to be constant.
First up in the results is the baseline electric potential variation in the different types of metal, which can help you determine which areas are more prone to corrosion. In the left image below, you can see that the applied current (100 A) causes a potential drop of about 2.5 mV over the busbar.
Next, you can examine the difference between the electric potential in the metal and the electrolyte film potential (shown below on the right as the variation in the electrode potential versus adjacent reference), which shows how atmospheric corrosion affects each type of metal. Notice how it's positive over the copper flange (red) and negative over the zinc nut and bolt (light blue) and aluminum flange (dark blue), indicating that the copper is more noble than the other materials, which are more likely to corrode in this model.
Electric potential variation in the busbar (left). Electric potential in the metals versus electrolyte film potential (right).
Now, let's look at the electrolyte potential variation in the electrolyte film, the results of which can be seen below. By examining this variation in the film covering the busbar, you can get an idea for where the reactions are likely to occur for the anodic and cathodic areas.
Electrolyte potential obtained in the electrolyte film covering the busbar.
To analyze the metal dissolution reaction, you can start by evaluating the local current for the density variation. As shown below, this reaction occurs mostly around the intersection between the copper flange and the zinc bolt (at the zinc surface) and also between the copper and aluminum flange at the aluminum surface. These results indicate that, as expected, the areas of localized corrosion occur between two dissimilar metals as well as the least noble metal.
The local current density for the metal dissolution electrode reaction over the exterior surfaces of the busbar.
In the results for the cathodic reaction, you can see that the oxygen reduction occurs on the aluminum and zinc surfaces. Here, it is clear that the transport of oxygen is limiting the corrosion process because the magnitude of the local oxygen reduction current density is close to the limiting current density.
The local current density for the oxygen reduction electrode reaction over the exterior surfaces of the busbar.
After studying the corrosion process and figuring out where each type of reaction is likely to occur, engineers can design busbars with better protection from atmospheric corrosion with various environmental conditions in mind.
To examine the atmospheric corrosion process yourself, you can click the button below to try out the busbar model featured here. Doing so will take you to the Application Gallery, which has documentation for this example and the related MPH-file (you will need a valid software license to download this file).


Some of the most penetrating words in all of Scriptures are found in the Old Testament Prophet, Haggai. They are penetrating because they have immediate relevant to every believer. And they touch us in some of the most carefully guarded places in our hearts.
Haggai was a prophet raised up by the Lord to accompany His people on their return to Judah and Jerusalem following the 70 long years of captivity in Babylon. And when they returned they faced the temple, God's house in ruin and in desperate need of rebuilding. But they like so many of us today have become completely preoccupied and engrossed with their own needs and situations. And the result as imply by Haggai probing questions is that, in spit of all of their effort and dedication without God's hand of blessing in their lives, they and we would never be satisfied regardless of how much material wealth or possession we acquired. These are questions that must be asked again and again among God's people.


text:
Historic Derbyshire farm to be turned into hotel and restaurant
Tim Paget
An historic farm in the Derbyshire countryside is set to gain a new lease of life as a sheep dairy, creamery, hotel, bar and restaurant.
The new business near Crich would look after 7,000 breeding ewes and 2,000 lambs.
Amber Valley borough councillors unanimously approved the plans, for Wakebridge Farm, north west of Crich, at a meeting last week.
The business, run by John Bailey and Heather Benbow, would make artisan cheese, including from sheep's milk.
Overall the scheme would create 20 to 30 jobs.
They hope to offer opportunities to the public, particularly children and people from non-rural backgrounds, to get involved and observe how products are made 'from field to plate'.
It also aims to provide agricultural and veterinary employment and training opportunities.
The application also states that the proposal would 'bring new agricultural technologies to Derbyshire which although present on the continent are rare in the UK'.
It would also have a 54-space car park and two timber cabins for dairy workers.
There was concern among local residents that the application would see the 'tranquility' of the parish 'destroyed' and that there was 'future potential for noisy functions'.
Borough councillor Gareth Gee, the ward member for the site, said that he had listened to the concerns of residents but approved of the application.
He said: "The residents are rightfully concerned with regard to the intended principle use of the farmstead being agricultural rather than as a hotel and function facilities.
"These concerns are borne out of the provision of a large car park and supplementary staff accommodation.
Heatwave predicted as temperatures set to reach 34C - but how long will it last?
"In addition to this I am told the applicant has recently undergone extensive road forming works to create a more permanent access to the rear of the property. These works have being carried out by heavy plant machinery and there is a concern over whether such works are allowable prior to the application being determined by the council.
"However, this is a positive move for Crich and Wakebridge, and I believe it is a local businessman who is reinvesting into the community. The proposals are quite impressive."
One of the owners of the farm, Mr Bailey, said that a great deal of work was required to one of the former buildings on the site, which is Grade II listed.
He told the meeting: "The roof is falling in, the floors are coming through and we need to spend a lot of money to protect this.
"This is a sustainable move and would be preferable to make it available for use by the public.
"We have come up with a plan to sort out this mess, and it would be for a sustainable agricultural use. We aim to restore the buildings and set up a restaurant and meeting room in some of the others.
"Our love is milking sheep and there is no danger of this causing any problems.
"The former use of the site was as a dairy for cows, and there is much less odour, much less noise and much less hassle from sheep.
"The sheep would have to be housed in winter, not left out on the hillside, because they are to be used for milk.
"There will be a gantry through the sheepshed for children and visitors to see the sheep up close.
"It will create 20-30 – some very highly-skilled – jobs, and opportunities for training," Mr Bailey added.
Rae Gee, the principal planning officer for Amber Valley Borough Council, said: "The previous purely agricultural use would not be able to bring all of the buildings back into use."
She urged councillors to support the application due to the economic benefit of the scheme, and to reopen derelict buildings.
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text:
\section{Introduction}

The classical Waring problem deals with expressing natural numbers as a sum of $k$th powers, where $k$ is a positive integer. Modern versions consider the same question, but over objects with non-commutative structures. For example, Shalev \cite{Sh} showed that for every finite (nonabelian) simple group of sufficiently high order every element can be expressed as values of word $w$ of length $3$. This was later improved to $2$ by Larsen, Shalev and Tiep  \cite{LST}. Larsen conjectured (in a personal communication) that a similar result should hold for matrices over finite fields. In other words, if $R$ denotes a commutative ring with unity, then the Matrix Waring Problem is to address whether matrices over $R$ can be expressed as a sum of two $k$th powers (of matrices). The goal of this article is to answer this question in the case where $R$ is a finite field $\mathbb{F}_q$, with $q$ sufficiently large; see Theorem \ref{intro_main_thm_1}.

This paper is a continuation of \cite{Ki}, which uses Lang-Weil's results on the number of solutions to equations over finite fields \cite{LW} to prove that for all integers $k \geq 1$, there exists a constant $C_k$ depending only on $k$ such that for all $q > C_k$ (i) for all $n = 1, 2$ every matrix in $M_n(\mathbb{F}_q)$ is a sum of two $k$th powers, and (ii) for all $n \geq 3$ every matrix in $M_n(\mathbb{F}_q)$ is a sum of \textit{at most} three powers. In this article, we strengthen this result and prove Larsen's conjecture:

\begin{thm}\label{intro_main_thm_1}
For all integers $k \geq 1$, there exists a constant $C_k$ depending only on $k$, such that for all $q > C_k$ and for all $k \geq 1$ every matrix in $M_n(\mathbb{F}_q)$ is a sum of two $k$th powers.
\end{thm}

A brief description of the strategy of the proof follows. Fix $k \geq 1$. Given a matrix  $C$ in $M_n(\mathbb{F}_q)$, the goal is to find matrices $A$ and $B$ in $M_n(\mathbb{F}_q)$ such that $C = A^k + B^k$. This may be viewed in the algebraic-geometric way as follows. The entries of $C$ are given in terms of homogeneous polynomials in $2 k^2$ variables of $A$ and $B$ so that the above equation defines an affine variety over $\mathbb{F}_q$ defined by $k^2$ polynomials in $2k^2$ variables, and the problem reduces to that of proving that the affine variety has an $\mathbb{F}_q$-rational point.  Let us recall Lang-Weil's theorem on the number of points of varieties over finite fields \cite{LW}. If an affine variety over $\mathbb{F}_q$ is geometrically irreducible and of dimension $n$ then the number of $\mathbb{F}_q$-rational points is given by $q^n + O(q^{(n-1)/2})$, where the implicit constant in the error term depends on the number and the degree of the equations defining $X$. Note that absolutely irreducible means that the affine variety remains irreducible when considered over the algebraic closure of $\mathbb{F}_q$. To solve our problem, we use a consequence of Lang-Weil's theorem found in \cite{Sc} to prove that the particular equation $X_1^{k} + \ldots + X_n^k = 1$ has a rational solution $(x_1, \ldots, x_n)$ which is `special' in the sense that the $x_i^k$ are nonzero and mutually distinct; the $k$th powers $x_i^k$  are used in constructing semisimple matrices that are $k$th powers.

We begin by observing that if a matrix is a sum of two $k$th powers, so is any of its conjugates. Furthermore, if each of the components of a direct sum of matrices is a sum of two $k$th powers so is their direct sum. These observations allow us to apply the theory of canonical forms of matrices and reduce the problem to that of expressing special Jordan matrices $J_{\alpha,n}$ associated to a primitive root $\alpha$ of $\mathbb{F}_q$ and the regular nilpotent matrices $J_{0,n}$ as a sum of two $k$th powers; see notation \S $2$. While the proof of expressing $J_{\alpha,n}$ as a sum of two $k$th powers is relatively easy, the proof of expressing nilpotent Jordan block $J_{0,n}$ takes some work.

With these results in hand we construct the matrix $A$ (in $J_{0,n} = A^k + B^k$) with $x_i^k$ as the eigenvalues in $\mathbb{F}_q$, so that $A$ is diagonalizable and is also a $k$th power. Based on $A$ the matrix $B$ though forced (so that the sum $A^k + B^k$ is equal to $J_{0,n}$) is not entirely so. By carefully choosing some entries of $B$ we ensure that it is also diagonalizable and is a $k$th power. Thus we prove that for every $k \geq 1$ and for every $n \geq 1$ there exists a constant $C(k,n)$ depending on \tit{both} $k$ and $n$ such that for all $q > C(k,n)$ the $J_{0,n}$ is a sum of two $k$th powers. But we want a constant depending only on $k$ but not on $n$. So, we give an independent proof that $J_{0,n}$ is a sum of two (both nilpotent this time) $k$th powers for all $n \geq 2k$. Precisely, we prove the following result which is of independent interest.

\begin{thm}\label{theorem-power-nilpotent}
Let $F$ be a field (not necessarily finite). Let $ k \geq 2$ and $n \geq 2k $. Then, $J_{0,n}$ (and hence every nilpotent matrix) is a sum of two $k$th powers in $M_n(F)$.
\end{thm}

By the first-proof of the representibility of $J_{0,n}$ we obtain finitely many constants $C(k,1), \ldots, C(k,2k-1)$. By Theorem \ref{theorem-power-nilpotent} we obtain a constant that works for all $n \geq 2k$. Thus, for all $q$ greater than the maximum of the \tit{finitely many} constants we obtain Theorem \ref{intro_main_thm_1} as required.

As for the non-nilpotent case, it is fairly straightforward to construct matrices $A$ and $B$ such that $J_{\alpha,n} = A^k + B^k$. Indeed we prove that the primitive element $\alpha$ can be expressed as $\alpha = a^k + b^k = c^k + d^k$ such that $a^k \neq c^k$ and $b^k \neq d^k$, and use this distinction to construct $A$ and $B$ which are, essentially, a direct sum of $2 \times 2$ matrices that are diagonalizable and are $k$th powers.

The paper is organized as follows. In \S \ref{Weil_results} we obtain consequences of the Weil's results on the number of solutions to equations over finite fields. In \S \ref{reduction} we reduce the problem to that of proving that the Jordan block associated to the primitive elements and the nilpotent Jordan block is a sum of two $k$th powers. Furthermore, we give quick proof of the non-nilpotent case. Sections  \S \ref{nilpotent_case_1} and \S \ref{nilpotent_case_2} form the essential content of this article. We prove $J_{0,n}$ is a sum of two $k$th powers that includes proving Theorem \ref{theorem-power-nilpotent}. In the final section \S \ref{main_result} we prove Theorem \ref{intro_main_thm_1} assembling various results in \cite{Ki}, \cite{Sm},  and the results  of the previous sections.

\section{Notation} \label{notation}
The $n \times n$ diagonal matrix with entries $\lambda_1, \ldots, \lambda_n$ along the diagonal is denoted by $\textrm{diag}(\lambda_1, \ldots, \lambda_n)$. If $A$ is an $r \times r$ matrix and $B$ is an $s \times s$ matrix, the direct sum $A \oplus B$ is the $(r + s) \times (r + s)$ block diagonal matrix  $ \begin{bmatrix} A & 0 \\ 0 & B \end{bmatrix}$. A finite field with $q$ elements is denoted by $\mathbb{F}_q$.

Let $F(X_1, \ldots, X_n)$ be a polynomial in $n$ indeterminates $X_1, \ldots, X_n$. An $n$-tuple $(x_1, \ldots, x_n) \in \mathbb{F}_q^n$ such that $F(x_1, \ldots, x_n) = 0$ is called a \textit{solution of $F$ in $\mathbb{F}_q^n$.}

Let $n \geq 1$ be a positive integer. Let $\lambda \in \mathbb{F}_q$. Let $J_{\lambda,n}$ denote ($J$ for Jordan) the following matrix:
\begin{equation}\label{Jordan_lambda_0}
J_{\lambda,n} =
\begin{bmatrix}
\lambda & 1 & 0 & \cdots  & 0 \\
0 & \lambda & 1 & \cdots  & 0\\
\vdots & \vdots & \ddots & \ddots & \vdots \\
0 & 0 & \cdots   & \lambda &1 \\
0 & 0 & \cdots    & 0& \lambda
\end{bmatrix}_{n \times n},
\end{equation}
where the diagonal entries are all $\lambda$, the super-diagonal entries are all $1$ and the remaining entries are $0$.

Let $n \geq 1$ and $r \geq 1$ be positive integers. Let $f := x^n -a_{n-1} x^{n-1}- \ldots - a_1 x - a_0 \in \mathbb{F}_q[x]$. The Jordan block $J_{f,r}$ associated to $f$ with $r$ blocks is matrix,
\begin{equation}\label{Jordan_f}
J_{f,r} :=
\begin{bmatrix}
C_{f} & I & 0 & \ldots & 0 &0 \\
0 & C_{f} & I & \ldots & 0 & 0\\
0 & 0 & C_{f} & \ldots & 0 & 0\\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & 0 & \ldots & C_{f} & I \\
0 & 0 & 0 & \ldots & 0 & C_{f}
\end{bmatrix}_{n r  \times n r}
C_{f} :=
\begin{bmatrix}
0 & 0 &  \ldots & 0 & a_0  \\
1 & 0 &  \ldots & 0 & a_1 \\
0 & 1 &  \ldots & 0 & a_2 \\
\vdots &\vdots  & \ddots &   & \vdots \\
0 & 0 & \cdots& 1 &  a_{n-1}
\end{bmatrix}_{n \times n}
\end{equation}
where $C(f)$ is the companion matrix of $f$, and $I$ is the $n \times n$ identity matrix.

\section{Equations over Finite Fields}\label{Weil_results}

Let $k \geq 1$ be a positive integer. In this section we prove that the equation $X_1^k + \ldots + X_n^k = c$, where $c$ is a nonzero element in $\mathbb{F}_q$, has sufficiently many $\mathbb{F}_q$-rational solutions, especially those solutions $(x_1, \ldots, x_n)$ whose $k$th powers $x_i^k$ are nonzero and mutually distinct. Eventually $x_i^k$ serve as eigenvalues of matrices we construct later.

We use Lang-Weil's result on the number of solutions to equations over finite fields \cite{LW}. For an introduction to solutions of equations over finite fields the reader may refer to \cite{Sc}.
\begin{thm}\label{weil_thm} (Weil)
Let $k > 0$ be a positive integer, and consider the following polynomial in $\mathbb{F}_q[X_1, \ldots, X_n]$:
$$
F(X_1, \ldots, X_n) := a_1X_1^{k} +  a_2 X_2^{k} + \ldots + a_n X_n^{k}- 1,
$$
where $a_i \neq 0$ for all $1 \leq i \leq n$. Let $N$ be the number of solutions  in $\mathbb{F}_q^n$ of the equation $F(X_1, \ldots, X_n) = 0$. Then
$$
\abs{N - q^{n-1}} \leq k^n \sqrt{q^{n-1}} \left( \frac{q}{q-1} \right)^{n/2}.
$$
Since $q/(q-1) \leq 2$, it follows that if $k \geq 2$ then
\begin{equation}\label{weil_inequality}
\abs{N - q^{n-1}} \leq k^{2n} \sqrt{q^{n-1}}.
\end{equation}
\end{thm}

\begin{proof}
For a proof the reader may refer to Theorem $5$A [page 160, \cite{Sc}]. Note that we modified the statement of Theorem $5$A to suit our needs; this modified statement is an immediate and a trivial consequence of Theorem $5$A.
For the last part, note that
$$
k^n \sqrt{q^{n-1}} \left( \frac{q}{q-1} \right)^{n/2} \leq k^n \sqrt{q^{n-1}} (\sqrt{2})^{n} \leq k^n \sqrt{q^{n-1}} k^n \leq k^{2n} \sqrt{q^{n-1}}.
$$
\end{proof}

Our goal is to find a solution $(x_1, \ldots, x_n)$ in $\mathbb{F}_q^n$ to $F$ such that the $k$th powers $x_i^k$ of the coordinates $x_i$ are  nonzero and mutually distinct

\begin{prop}\label{weil_nilpotent_case}
Let $c$ be any nonzero element in $\mathbb{F}_q$. Fix positive integers $n \geq 3$ and $k \geq 1$, and consider the polynomial
$$
F(X_1, \ldots, X_n) := X_1^{k} + X_2^{k} + \ldots + X_n^k - c \in  \mathbb{F}_q[X_1, \ldots, X_n].
$$
There exists a constant $C(k,n) :=$ {\normalfont max} $(2n^2, k^{4n/(n-8)})$ (which depends on both $k$ and $n$) such that for all $q >  C(k,n)$ there exists a solution $(x_1, \ldots, x_n)$ to $F = 0$ in $\mathbb{F}_q^n$ such that
\begin{enumerate}[(a)]
\item \label{nonzero}
$x_i \neq 0$, for all $1 \leq i \leq n$;
\item \label{mutually_distinct}
$x_i^k \neq x_j^k$, for all $1 \leq i , j \leq n$ and $i \neq j$.

\end{enumerate}
\end{prop}

\begin{proof} Let $p$ be the characteristic of $\mathbb{F}_q$. Let $k = l \cdot p$ for some integer $l \geq 0$. The $p$-th power map on $\mathbb{F}_q$ is an automorphism of $\mathbb{F}_q$, so the equation $X_1^l + \ldots + X_n^l - c = 0$ has a solution satisfying the conditions \eqref{nonzero} and \eqref{mutually_distinct} if and only if its $p$-th power
$$
(X_1^l + \ldots + X_n^l - c)^p = X_1^k + X_2^k + \ldots + X_n^k - c^p,
$$
has a solution satisfying the same conditions. Hence we may assume that $p \nmid k$.
Let $N_0$ be the number of solutions $(x_1, \ldots, x_n)$ in $\mathbb{F}_q^n$  to  the equation $F(X_1, \ldots, X_n) = 0$. By Theorem \ref{weil_thm},
$$
q^{n-1} - C_0 \leq N_0 \leq q^{n-1} + C_0, \trm{ where }C_0 = k^{2n} \sqrt{q^{n-1}} .
$$

Let $N_1$ be the number of solutions $(x_1, \ldots, x_n)$ to the equation $F=0$  in $\mathbb{F}_q^n$ with $
x_{i}^k = x_{j}^k $ for some $i,j$. Then $(x_1, \ldots , x_i, \ldots, \widehat{x_j}, \ldots x_n)$ is a solution to the equation
\begin{align*}
&X_{1}^k + \ldots  + 2 X_{i}^k + \ldots + \widehat{X_{j}^k} + \ldots + X_n^k = 1.
\end{align*}
where the $\widehat{\cdot}$ denotes that the term is dropped. By hypothesis that the characteristic of $\mathbb{F}_q$ is not $2$, so the above equation is an equation in $n-1$ indeterminates $X_1, \ldots, X_i, \ldots, \widehat{X_j}, \ldots, X_n$. Taking into account that there are ${n \choose 2}$ such pairs $(i,j)$, it follows from Theorem \ref{weil_thm} that $N_1$ satisfies the following inequality:
$$
{n \choose 2} \left(q^{n-2} - C_1 \right) \leq N_1  \leq {n \choose 2} \left( q^{n-2} +  C_1 \right),
$$
where $C_1 = k^{2(n-1)} \sqrt{q^{n-2}} \leq k^{2n} \sqrt{q^{n-1}}$.
Similarly, let $N_2$ be the number of solutions $(x_1, \ldots, x_n)$ to the equation $F(X_1, \ldots, X_n)=0$ in $\mathbb{F}_q^n$ with $
x_i = 0 $  for some $1 \leq i \leq n$ . Then
$(x_1, \ldots, \widehat{x_i}, \ldots x_n)$,
is a solution to the equation
\begin{align*}
&X_{1}^k + \ldots  +  X_{i-1}^k + X_{i+1}^k + \ldots + X_n^k = 1.
\end{align*}
Again by Theorem \ref{weil_thm}, $N_2$ satisfies the inequality
$$
{n \choose 1} \left( q^{n-2} - C_1 \right) \leq  N_2 \leq {n \choose 1}  \left(  q^{n-2} + C_1 \right)
$$
where $C_1$ is as above. Note that $N_0 - N_1 - N_2$ is \textit{not} the exact number of solutions satisfying the desired conditions, because there are solutions counted in $N_1$ and $N_2$, but the exact number of solutions is at least $N_0 - N_1 - N_2$. Therefore, we may find a lower bound for $N_0 - N_1 - N_2$, and one such bound is given by
\begin{align*}
&\left(q^{n-1} - C_0 \right) - {n \choose 2} \left(q^{n-2} + C_1 \right) - {n \choose 1}\left( q^{n-2} + C_1 \right)\\
&= \left( q^{n-1} -  {n +1 \choose 2} q^{n-2} \right)  - \left( C_0 +{n +1 \choose 2}  C_1 \right) \\
& = q^{n-2} \left( q  - {n +1 \choose 2} \right)- \left( C_0 +{n +1 \choose 2}  C_1 \right)
\end{align*}
Let $x$ denote the first summand $q^{n-2} \left( q  - {n +1 \choose 2} \right)$ and $y$ denote the second summand $C_0 +{n +1 \choose 2}  C_1 $.
To find a lower bound for the above expression, we need to find an upper bound on $y$ for fixed $k$ and $n$. For the first term, note that since $2n^2  \geq {n+1 \choose 2}$
we require $q > 2n^2$.
With this choice of $q$, an upper bound of the second term is obtained using the bounds on $C_0$ and $C_1$ obtained above:
\begin{align*}
y&\leq  k^{2n} \sqrt{q^n}+ {n +1 \choose 2} k^{2n} \sqrt{q^{n-1}} \leq k^{2n} q^{n/2} (1 + 2n^2)\\ &\leq k^{2n} q^{n/2}(1 + q) \leq k^{2n} q^{(n/2) + 2}
\end{align*}
Since we want $x > y$, it suffices to choose $q$ such that $q > 2n^2$ and $q^{n-2} \geq  k^{2n} q^{(n/2) + 2}$. From the second inequality it follows that we require $q^{n-8} > k^{4n}$ and so it suffices to have $q > k^{4n/(n-8)}$. Thus it suffices to define $C(k,n) := \textrm{max} \left( 2n^2,  k^{4n/(n-8)} \right)$.
\end{proof}

\noindent The special case of Theorem \ref{weil_thm} with $n =2$ is required to write the Jordan block $J_{\alpha, n}$ where $\alpha$ is a primitive element of $\mathbb{F}_q$ as a sum of two $k$th powers. We will see later that two solutions $(a,b)$ and $(c,d)$ in $\mathbb{F}_q \times \mathbb{F}_q$ are required to $X_1^k + X_2^k = \lambda$ such that $a^k \neq c^k$ and $b^k \neq d^k$ for any fixed $k \geq 2$.

\begin{prop}\label{non_nilpotent_case}
Let $k \geq 2$, and the characteristic of $\mathbb{F}_q$ not $2$. Let $\lambda$ be a nonzero element in $\mathbb{F}_q$. Consider the equation
\begin{equation}\label{2_equation}
X_1^k  + X_2^k = \lambda.
\end{equation}
For all  $q > k^{16}$, there exists two solutions $(a,b)$ and $(c,d)$  to \ref{2_equation} in $\mathbb{F}_q \times \mathbb{F}_q$ such that $a^k \neq c^k$ and $b^k \neq d^k$.
\end{prop}

\begin{proof}
Let $a_1 =1/\lambda$ and $a_2 = 1/\lambda$ in Theorem \ref{weil_thm}, then the number of solutions $N$ to the equation \eqref{2_equation} satisfies the following inequality
$
\abs{N - q} \leq k^4 \sqrt{q}.
$
 The number of solutions with fixed $X_1^k \in \mathbb{F}_q$ is at most $k^2$ and similarly with fixed $X_2^k$ is at most $k^2$. Thus for a pair of solutions $(a,b)$ and $(c,d)$ with $a^k \neq c^k$ and $b^k \neq d^k$ to exist we require $q - k^4 \sqrt{q} \geq 2k^2 + 1$. Since $k \geq 2$ it suffices to have $\sqrt{q} - k^4 > 0$ and $\sqrt{q} \geq k^4 \geq 2k^2 +1$. Thus if $q > k^{16}$ we have the existence of two solutions with the desired properties.
\end{proof}

\section{Reducing the Problem}\label{reduction}

We begin by reducing the problem to that of representing two special matrices as a sum of two $k$th powers. Before that, let us recall the following elementary fact. Let $R$ and $S$ be rings and $\phi \colon  R  \to S$ be a ring homomorphism. For any integer $n \geq 1$, the canonical map $\Phi \colon M_n(R) \to M_n(S)$ induced by $\phi$ as $[a_{ij}] \mapsto [\phi(a_{ij})]$ is also a ring homomorphism, and injective if $\phi$ is injective. Also, recall that for all integers $m, n \geq 1$ the ring $M_m(M_n(R))$ is canonically isomorphic to the ring $M_{mn}(R)$.

We are interested in the special case where $R = \mathbb{F}_{q^n} = \mathbb{F}_q(\alpha)$ and $S = M_n(\mathbb{F}_q)$, where $\alpha$ is a primitive element of $R$.  Consider the ordered basis $(1, \alpha, \ldots, \alpha^{n-1})$ of $R$ over $\mathbb{F}_q$. For $\beta \in R$, the multiplication-by-$\beta$ map $ \phi_{\beta}\colon x \mapsto \beta \cdot x$ is an $\mathbb{F}_q$-linear map, represented in the above basis by a matrix in $S$ giving the map $\phi\colon R \to S$. In particular, if $f(t) = t^n -a_{n-1} t^{n-1}- \ldots - a_1 t - a_0$ is minimal polynomial of $\alpha$ then $\alpha$ is mapped to $C_f$. Together with the canonical isomorphism  $M_{nd}(\mathbb{F}_q) \cong M_d(M_n(\mathbb{F}_q))$ we obtain a canonical embedding  $M_d(R) \hookrightarrow M_{nd}(\mathbb{F}_q)$ that maps $J_{\alpha,d}$ to $J_{f,d}$; see \S 2 for notation.

\begin{prop} \label{reduction}
Fix an integer $C \geq 0$.  Let $k \geq 1$ be a positive integer. For all $q > C$, $n \geq 1$ and $\alpha$ any primitive element of $\mathbb{F}_q$, suppose that matrices  $J_{\alpha, n}$ and $J_{0,n}$ in $M_n(\mathbb{F}_q)$ are a sum of two $k$th powers. Then, for all $q > C$ and $n \geq 1$, every matrix in $M_n(\mathbb{F}_q)$ is a sum of two $k$th powers.
\end{prop}

\begin{proof}

Let $A \in M_n(\mathbb{F}_q)$. Then $A$ is a sum of two $k$th powers if, and only if, for all $P \in GL_n(\mathbb{F}_q)$ the conjugate $P A P^{-1}$ of $A$ is a sum of two $k$th powers; see Lemma $3.1$ of \cite{Ki}. Therefore, it suffices to prove that  some, hence any, choice of representative of the conjugacy classes of $A$ is a sum of two $k$th powers. It is well known that if $f(t)$ is the characteristic polynomial of $A$ with the factorization $\prod_{i =1}^k f_i(t)^{r_i}$, where $r_i \geq 1$ and $f_i(t)$ are irreducible polynomials over $\mathbb{F}_q$, then $A$ is conjugate to the block diagonal matrix $J_{f_1, r_1} \oplus  \ldots \oplus J_{f_k,r_k}$. Clearly, if each Jordan block $J_{f_i, r_i}$ is a sum of two $k$th powers, then so is their direct sum. Therefore, it suffices to prove that matrices of the form $J_{f,r}$ is a sum of two $k$th powers, where $r \geq 1$ and $f$ is, by abuse of notation, an irreducible polynomial over $\mathbb{F}_q$. If $f(t) = t$ with zero constant term then $J_{f, r}$ is the regular nilpotent matrix of size $r$, and it is a sum of two $k$th powers by hypothesis. On the other hand, by the preceding discussion before the theorem, the matrix $J_{f, r}$ in the case where $f(t) \neq t$ is irreducible over $\mathbb{F}_q$ and $r \geq 1$,  it is a sum of two $k$th powers if $J_{\alpha,r}$ is a sum of two $k$th powers, which indeed is the case by hypothesis.
\end{proof}

Thus we are reduced to showing that $n \times n$ matrices $J_{\alpha,n}$, $J_{0,n}$  are a sum of two $k$th powers. While the former case is almost immediate, it is the nilpotent case that takes some work.

First we prove that the Jordan cell of the form $J_{\alpha,n}$ where $\alpha$ is a primitive element of $\mathbb{F}_q$ is a sum of two $k$th powers, for all sufficiently large $q$. The reader may notice that the following proposition holds for all $n \geq 2$, in particular including $n = 2$. So the result includes some of the results in \cite{Ki}. But not everything found in \cite{Ki} is subsumed by the results in this article. We will see that we still need results in \cite{Ki} in the characteristic $2$ case.

\begin{prop}\label{primitive_element}
For every integer $k \geq 1$ there exists a constant $C_k$ depending only on $k$ such that for all $q > C_k$ and $n \geq 2$ every matrix $J_{\alpha,n}$ is a sum of two $k$th powers,  where $\alpha$ is a primitive element of $\mathbb{F}_q$.
\end{prop}
\begin{proof}

By Proposition \ref{non_nilpotent_case}, given $k \geq 2$ there exists a constant $C_k$ depending only on $k$ such that for all $q > C_k$ there exists pairs $(a,b)$ and $(c,d)$ in $\mathbb{F}_q \times \mathbb{F}_q$ such that $a^k \neq c^k$ and $b^k \neq d^k$ and $a^k + b^k = c^k + d^k = \alpha$. For these two $2$-tuples we define the following matrices.

For $n \geq 2$ even, let
$$
G_n = \bigoplus_{n /2} \begin{bmatrix} a^k &  1\\ 0 & c^k \end{bmatrix} \trm{ and } H_n = [b^k] \oplus \left( \bigoplus_{(n-2)/2} \begin{bmatrix} d^k & 1 \\ 0 &b^k \end{bmatrix} \right) \oplus [d^k].
$$
For $n \geq 3$ odd, let
$$
G_n = \left( \bigoplus_{(n-1)/2} \begin{bmatrix} a^k & 1 \\ 0 & c^k \end{bmatrix} \right) \oplus [a^k] \trm{ and } H_n = [b^k] \bigoplus \left( \bigoplus_{(n-1)/2} \begin{bmatrix} d^k & 1 \\ 0 & b^k \end{bmatrix} \right).
$$
For example, if $n =5$ the matrices $G_5$ and $H_5$ are of the following form:
$$
G_5 =
\begin{bmatrix}
a^k & 1 &  &  &  \\
  & c^k  &  &   &  \\
& & a^k & 1 &  \\
& & & c^k &  \\
& & & & a^k \\
\end{bmatrix}
\textrm{ and }
H_5 =
\begin{bmatrix}
b^k &  & & & \\
  & d^k  &  1& &  \\
& &   b^k &   & \\
& & &  d^k & 1\\
& & &  & b^k \\
\end{bmatrix}.
$$
Clearly $\begin{bmatrix} a^k & 1 \\  & c^k \end{bmatrix}$ is diagonalizable because it has distinct eigenvalues $a^k$ and $c^k$, so it is conjugate to $\begin{bmatrix} a^k &  \\  & c^k \end{bmatrix}$, visibly a $k$th power. Similarly $\begin{bmatrix} d^k & 1 \\  & b^k \end{bmatrix}$  is diagonalizable because it has distinct eigenvalues $d^k$ and $b^k$, so it is conjugate to a diagonal matrix $\begin{bmatrix} d^k &  \\  & b^k \end{bmatrix}$ which is $k$th power too. Since representation of a matrix as a sum of two $k$th powers is stable under direct sum of matrices, it follows that $G_n$ and $H_n$ are $k$th power and therefore their sum is represented as a sum of two $k$th powers for all $q > C_k$.
\end{proof}
We consider the remaining and more difficult case in the next section, namely that of representing the nilpotent matrix $J_{0,n}$ as a sum of two $k$th powers.

\section{Nilpotent Case- I}\label{nilpotent_case_1}

We being by proving the following \tit{weaker} result. Note that the constant given by this result depends on \textit{both} $n$ and $k$, while we desire to have a constant depending only on $k$ which we will achieve in the next section.
\begin{prop}\label{C_depend_k_n}
For every integer $k \geq 1$ and for every $n \geq 3$ there exists a constant $C(k,n)$ depending on both $k$ and $n$ such that for all $q > C(k,n)$ with the characteristic of $\mathbb{F}_q$ not $2$, the matrix $J_{0,n}$ is a sum of two $k$th powers where at least one of them is diagonalizable.
\end{prop}

\begin{proof}
For a given $k \geq 1$ and $n \geq 3$ define matrices $A_n$ and $B_n$ as follows: let $y_1, \ldots, y_{n-1}$ and $z_1, \ldots, z_{n-1}$ be elements in $\mathbb{F}_q$ which will be chosen later. Define $A_n = [a_{ij}]$ where
$$
a_{ij} = \begin{cases}
1 & \textrm{ if } j = i +1, \; \;  1 \leq i \leq n-1 \; \\
1 & \textrm{ if } i = j = n \; \\
y_j & \textrm{ if } i = n, \; \; 1 \leq j \leq n-1  \; \\
0 & \textrm{ otherwise }
\end{cases}.
$$
In other words the last row of $A$ is $[y_1 \; y_2 \; \ldots \; y_{n-1} \; 1]$, with $1$'s immediately above the diagonal (also called super-diagonal) and $0$s everywhere else.
Define $B_n = [b_{ij}]$ where
$$
b_{ij} = \begin{cases}
1 & \textrm{ if } j = i+1, \; \; 1 \leq i \leq n-2 \\
-1 & \textrm{ if } i = j = n \\
z_i & \textrm{ if } j = n; \; 1 \leq i \leq n-2 \\
-y_j & \textrm{ if } i = n, \; \; 1 \leq j \leq n-1 \\
0 & \textrm{ otherwise }
\end{cases}.
$$
In other words, the last row of $B$ is $[-y_1 \; -y_2 \; \ldots \; -1 ]$ and the last column is $[z_1 \; z_2 \; \ldots \; 0 \; -1]^T$ with $0$ in the $((n-1), n)$ entry and $-1$ in the $(n,n)$ entry and $1$s along the super-diagonal except at  $((n-1), n)$-th position.
The sum of $A_n$ and $B_n$ is the matrix $(c_{ij})$ where
$$
c_{ij} = \begin{cases}
2 & \textrm{ if } j = i+1, \; \; 1 \leq i \leq n-2 \\
z_i & \textrm{ if } j = n, 1; \; 1 \leq i \leq n-2 \\
1	& \textrm{ if } i = n-1, j = n \\
0 & \textrm{ otherwise }
\end{cases}.
$$
For example, if $n = 4$ then
$$
A_4 = \begin{bmatrix}
0 & 1 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
y_1 & y_2 & y_3 & 1 \\
\end{bmatrix} \; \;
B_4 = \begin{bmatrix}
0 & 1 & 0 & z_1 \\
0 & 0 & 1 & z_2 \\
0 & 0 & 0 & 0 \\
-y_1 & -y_2 & - y_3 & -1 \\
\end{bmatrix}
\; \; A_4 + B_4 = \begin{bmatrix}
0 & 2 & 0 & z_1 \\
0 & 0 & 2 & z_2 \\
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 \\
\end{bmatrix}.
$$
By hypothesis the characteristic of $\mathbb{F}_q$ is not $2$, so the matrix $(c_{ij})$ is conjugate to $J_{0,n}$.  Therefore, it suffices to show that $A_n$ and $B_n$ are $k$th powers for appropriate choices of $y_i$ and $z_j$.

First we show that $A_n$ is a $k$th power for an appropriate choices of $y_i$. It is easy to prove by induction on $n$ that the characteristic polynomial of $A_n$ is
$$
t^n - t^{n-1} - y_{n-1} t^{n-2} - y_{n-2} t^{n-3} - \ldots - y_2 t - y_1.
$$
By Proposition \ref{weil_nilpotent_case}, there exists a solution $(\alpha_1, \ldots, \alpha_n) \in \mathbb{F}_q^n$ to $X_1^k + \ldots + X_n^k = 1$ such that $\alpha_i \neq 0$ for all $1 \leq i \leq n$ and $\alpha_i^k \neq \alpha_j^k$ for all $1 \leq i, j \leq n$ and $i \neq j$.
We want the characteristic polynomial of $A_n$ be equal to the polynomial
$$
(t - \alpha_1^k) \cdots (t - \alpha_n^k)
$$
so that we \textit{have} to define $y_j := (-1)^{n - j} S_{n-j+1}(\alpha_1^k, \ldots, \alpha_n^k)$, the $j$-th elementary symmetric polynomial in $\alpha_j^k$. Running the argument backwards, we see that with these choices of $y_j$, the characteristic polynomial of $A_n$ is
$$
t^n - t^{n-1} - y_{n-1} t^{n-2} - \ldots - y_1  = (t - \alpha_1^k) \cdots (t - \alpha_n^k).
$$
On the other hand, since $\alpha_i^k \neq \alpha_j^k$ for all $i \leq j$,  it follows that $A_n$ has $n$ mutually distinct eigenvalues in $\mathbb{F}_q$, so it is conjugate to the diagonal matrix $\textrm{diag}(\alpha_1^k, \ldots, \alpha_n^k)$ which is clearly a $k$th power. Therefore $A_n$ is conjugate to $k$th power, hence it is a $k$th power too.

Now we show that $B_n$ is a $k$th power for an appropriate choice of $z_i$s. The characteristic polynomial of $B_n$ is
\begin{align*}
&t^n + t^{n-1} \\
&+ (y_1 z_1 + y_2 z_2 + \ldots + y_{n-2} z_{n-2}) t^{n-2}  \\
&+ (y_1 z_2 + y_2 z_3 + \ldots + y_{n-3} z_{n-2}) t^{n-3} \\
&+ \ldots \\
&+ (y_1 z_{n-3}  + y_2 z_{n-2} ) t^{2} \\
&+ (y_1 z_{n-2} ) t.
\end{align*}
For the above choice of $k$ and $n$, again by Proposition \eqref{weil_nilpotent_case} there exists a constant $C(n-1,k)$ depending on both $n$ and $k$ such that for all $q > C(n-1,k)$ there exists an $(n-1)$-tuple $(\beta_1, \ldots, \beta_{n-1}) \in \mathbb{F}_q^{n-1}$ such that $(\beta_1, \ldots, \beta_{n-1})$ is a solution to
$$
X_1^k + \ldots + X_{n-1}^k = -1.
$$
Moreover, $\beta_i^k \neq \beta_j^k$ for all $1 \leq i, j \leq n-1$ and $i \neq j$, and $\beta_i \neq 0$ for all $1 \leq i \leq n-1$. Thus have a solution $(0, \beta_1, \ldots, \beta_{n-1}) \in \mathbb{F}_q^n$ such that the $k$th powers of the coordinates are mutually distinct and $\beta_i$ are nonzero.
Now, we force the characteristic polynomial of $B_n$ to be
$$
t (t - \beta_1^k) \cdots (t - \beta_{n-1}^k).
$$
Let $s_i$ denote the $i$th elementary symmetric polynomial in $\beta_1^k, \ldots, \beta_{n-1}^k$. Equating the coefficients, we obtain the following relations
\begin{align*}
y_1 z_1 + y_2 z_2 +y_3 z_3  \ldots  + y_{n-2} z_{n-2} &= s_2 \\
y_1 z_2 + y_2 z_3 + \ldots + y_{n-2} z_{n-2}  & =- s_3 \\
\vdots & \\
y_1 z_{n-3} + y_2 z_{n-2} &= (-1)^{n-2} s_{n-2} \\
y_1 z_{n-2} &= (-1)^{n-1}s_{n-1}.
\end{align*}
The coefficient matrix of the linear system above has nonzero determinant, in fact equal to $y_1^{n-1}$ (here is where we use that $y_1$ is nonzero), so it has a unique solution $[z_1 \; \ldots \; z_{n-1}]^{T}$. With these choices of $z_i$, it follows that the characteristic polynomial of $B_n$ is equal to $t(t - \beta_1^k) \cdots (t - \beta_{n-1}^k)$. Since the $k$th powers of $\beta_i$ are mutually distinct and nonzero it follows that, including the $0$ as an eigenvalue, $B_n$ has $n$ mutually distinct eigenvalues in $\mathbb{F}_q$, so it  is conjugate to the diagonal matrix $\textrm{diag}(0, \beta_1^k, \ldots, \beta_{n-1}^k)$ which is visibly a $k$th power.
\end{proof}

We reiterate for the sake of clarity that the above Proposition \ref{C_depend_k_n} is distinct from Theorem \ref{intro_main_thm_1}; the constant $C(k,n)$ depends on both $k$ and $n$, while the constant $C_k$ in Theorem \ref{intro_main_thm_1} depends only on $k$. We now prove that for all $n$ greater than a constant depending on $k$,  the  $J_{0,n}$ is a sum of two $k$th powers. The proof is independent of Proposition \ref{C_depend_k_n}. So we are not actually giving two proofs of the same result, the constant in Proposition \ref{C_depend_k_n} depends on $k$ and $n$ while the following result is one about representibility of $J_{0,n}$ for all $n \geq 2k$. Based on these two results we obtain a constant that is independent of $n$ but depends only on $k$.

\section{Nilpotent Case-II}\label{nilpotent_case_2}

Results in this section are valid for all fields, not necessarily finite. Let $F$ be any field and  $M_n(F)$ be the ring of $n\times n$ matrices with entries in $F$. Let $k \geq 2$ be a positive integer. Our goal is to express $J_{0,n}$ as a sum of two $k$th powers. In contrast with Proposition \ref{C_depend_k_n} which expresses the regular nilpotent matrix $J_{0,n}$ as a sum of $k$th powers of \textit{semisimple} elements of $M_{n}(\mathbb{F}_q)$, the main result of this section expresses $J_{0,n}$ as a sum of two $k$th powers of \textit{nilpotent} matrices. By Proposition \ref{reduction} it suffices to consider the regular nilpotent matrix $J_{0,n}$.

For a real number let $\lfloor x \rfloor$ denote the largest integer less than or equal to $x $ and $\lceil x\rceil$ denotes the smallest integer greater than or equal to $x$. Consider Lemma $8.1$ of \cite{KS} attributed to Miller and quoted here for sake of clarity.
\begin{lem} \label{miller}
(Miller) Let $n,r \geq 1$ be positive integers, and suppose $n > r$. Let $m$ be $n$ congruent modulo $r$ with $0 \leq m \leq n-1$. Then the $r$-th power $J_{0,n}^r$ is conjugate to
$$\left( \bigoplus_{r-m} J_{0,\lfloor\frac{n}{r}\rfloor} \right) \bigoplus \left( \bigoplus_{m} J_{0,\lceil\frac{n}{r}\rceil}\right).$$
\end{lem}

\begin{defn}
For a positive integer $n \geq 1$, the $r$-tuple $(n_1, \ldots, n_r)$, for $r \geq 1$, is a partition of $n$ if $1 \leq n_1 \leq \ldots \leq n_r$ and $n = n_1 + \ldots + n_r$. The coordinates $n_i$ are called {\normalfont parts of the partition}. The {\normalfont Junction matrix} associated to the partition $(n_1, \ldots, n_r)$ is the matrix $$
\mathfrak J_{(n_1, \ldots, n_r)} = e_{n_1, n_1+1}+ e_{(n_1+n_2), (n_1+n_2+1)}+\cdots + e_{(n_1+\ldots+n_{r-1}), (n_1+\ldots+n_{r-1}+1)}
$$
where $e_{i,j}$ denotes the matrix with $1$ at ij-th place and $0$ elsewhere.
\end{defn}
For example, corresponding to the partition $(1,1,2,2)$ and $(2,2,2)$ of $6$, the junction matrices are
$$\newcommand*{\multicolumn{1}{c|}{0}}{\multicolumn{1}{c|}{0}}
\left[\begin{array}{cccccc}
\multicolumn{1}{c|}{0} & 1 & 0 & 0 &0 &0  \\ \cline{1-2}
\multicolumn{1}{c|}{0} & \multicolumn{1}{c|}{0} &1 & 0 &0 &0 \\ \cline{2-4}
0&\multicolumn{1}{c|}{0}&0&\multicolumn{1}{c|}{0} &0&0\\
0 & \multicolumn{1}{c|}{0} & 0& \multicolumn{1}{c|}{0} & 1 &0 \\ \cline{3-6}
0 & 0& 0& \multicolumn{1}{c|}{0} & 0&0 \\
0 & 0& 0& \multicolumn{1}{c|}{0}& 0&0
\end{array}\right], \left[\begin{array}{cccccc}
0&\multicolumn{1}{c|}{0} & 0 & 0 &0 &0  \\
0 & \multicolumn{1}{c|}{0} &1 & 0 &0 &0 \\ \cline{1-4}
0&\multicolumn{1}{c|}{0}&0&\multicolumn{1}{c|}{0} &0&0\\
0 & \multicolumn{1}{c|}{0} & 0& \multicolumn{1}{c|}{0} & 1 &0 \\ \cline{3-6}
0 & 0& 0& \multicolumn{1}{c|}{0} & 0&0 \\
0 & 0& 0& \multicolumn{1}{c|}{0}& 0&0
\end{array}\right] .$$

Clearly the junction matrices are nilpotent. When these are $k$th power is determined as follows:
\begin{lem}\label{Lemma-junction-conjugate}
For a positive integer $k \geq 1$, let $(n_1, \ldots, n_k)$ be a partition of $n$. Suppose $n \geq 2k$ and $n_i \geq 2$ for all $1 \leq i \leq r$. Then $\mathfrak J_{(n_1, \ldots, n_k)}$ is a $k$th power.
\end{lem}
\begin{proof}
The junction matrix $\mathfrak J_{(n_1, n_2, \ldots, n_k)}$ associated to the partition $(n_1, \ldots, n_k)$ may be viewed as the matrix of the endomorphism $T$ of $F^n$ in the standard basis, say $(e_1, \ldots, e_n)$; in particular, it maps $e_{(n_1+\ldots + n_i)+1}$ to $e_{n_1+\ldots + n_i}$ for all $1 \leq i \leq r$ and maps other basis vectors to $0$. Consider the following basis obtained by reordering $(e_1, \ldots, e_n)$:
\begin{eqnarray*}
&(e_1,\ldots, e_{n_1-1}, e_{n_1+2}, \ldots, e_{(n_1+n_2-1)}, e_{(n_1+n_2+2)}, \ldots, \\ & e_{(n_1+\ldots +n_{k-1})-1}, e_{(n_1+\ldots+n_{k-1})+2},  \ldots, e_{(n_1+n_2+\ldots + n_k)}, \\&
 e_{n_1},  e_{n_1+1}, e_{n_1+n_2}, e_{(n_1+n_2 +1)} \ldots, e_{(n_1+\ldots +n_{k-1})}, e_{(n_1+\ldots+n_{k-1})+1}).
\end{eqnarray*}
Here we put the tuple of basis vectors $e_{n_1+\ldots +n_{i}}, e_{n_1+\ldots+n_{i}+1}$ for $1\leq i \leq r-1$ at the end. The matrix of $T$ in this basis is
$$
A:=\displaystyle \left( \bigoplus_{n-2(k-1)} J_{0,1} \right) \bigoplus \left(\bigoplus_{k-1} J_{0,2} \right),
$$
where $J_{0,1}$ is the $1 \times 1$ matrix $[0]$.
Now we use the hypothesis $n \geq 2r$; let
$$
B :=  \left( \bigoplus_{n-2(k-1)} J_{0,1} \right)  \bigoplus  J_{0,2(k-1)}.
$$
By Lemma~\ref{miller}, $B^k$ is conjugate to $A$ which in turn is conjugate to $\mathfrak J_{(n_1, n_2, \ldots, n_k)}$ so it is an $k$th power.
\end{proof}
The condition $n_i \geq 2$ in Lemma \ref{Lemma-junction-conjugate} is important. Indeed, if some $n_i < 2$ then junction matrix has a Jordan block of size $> 2$ so that it may not be a $k$th power. As an illustration of the above result, the junction matrix associated to the partition $(2,2,2,3)$ and its conjugate $A$ (where $A$ is as defined in the proof) are $\mathfrak J_{(2,2,2,3)}=$
$$ \newcommand*{\multicolumn{1}{c|}{0}}{\multicolumn{1}{c|}{0}}
\newcommand*{\multicolumn{1}{c|}{1}}{\multicolumn{1}{c|}{1}}
\left[\begin{array}{ccccccccc}
0&\multicolumn{1}{c|}{0}  & 0 & 0 &0 &0 &0&0&0 \\
0&\multicolumn{1}{c|}{0}  &1 & 0 &0 &0 &0&0&0\\ \cline{1-4}
0&\multicolumn{1}{c|}{0}&0&\multicolumn{1}{c|}{0} &0&0&0&0&0\\
0 & \multicolumn{1}{c|}{0} & 0& \multicolumn{1}{c|}{0} & 1 &0&0&0&0 \\ \cline{3-6}
0&0&0&\multicolumn{1}{c|}{0} &0&\multicolumn{1}{c|}{0}&0&0&0\\
0 & 0 & 0& \multicolumn{1}{c|}{0} & 0 &\multicolumn{1}{c|}{0}&1&0&0 \\ \cline{5-9}
0 & 0& 0& 0& 0&\multicolumn{1}{c|}{0}&0&0&0\\
0 & 0& 0& 0& 0&\multicolumn{1}{c|}{0}&0&0&0\\
0 & 0& 0& 0& 0&\multicolumn{1}{c|}{0} &0&0&0
\end{array}\right], A= \left[\begin{array}{ccccccccc}
0&0  & \multicolumn{1}{c|}{0} & 0 &0 &0 &0&0&0 \\
0&0  &\multicolumn{1}{c|}{0} & 0 &0 &0 &0&0&0\\
0&0&\multicolumn{1}{c|}{0} & 0&0&0&0&0&0\\\cline{1-5}
0 & 0 & \multicolumn{1}{c|}{0}&0 & \multicolumn{1}{c|}{1} &0&0&0&0 \\
0&0&\multicolumn{1}{c|}{0} &0&\multicolumn{1}{c|}{0}&0&0&0&0\\\cline{4-7}
0 & 0 & 0& 0 & \multicolumn{1}{c|}{0} &0&\multicolumn{1}{c|}{1}&0&0 \\
0 & 0& 0& 0& \multicolumn{1}{c|}{0} &0&\multicolumn{1}{c|}{0}&0&0\\\cline{6-9}
0 & 0& 0& 0& 0&0&\multicolumn{1}{c|}{0} &0&1\\
0 & 0& 0& 0& 0&0&\multicolumn{1}{c|}{0}&0&0
\end{array}\right].$$

Now we prove the main theorem of this section.

\begin{proof}
(of Theorem \ref{theorem-power-nilpotent}) We are given $k\geq 2$ and $k\geq 2r$. Let $m$ be the integer such that $ m \equiv n \pmod{k}$ and $0 \leq m < k$. Let
$$
A :=  \left( \bigoplus_{k-m} J_{0,\lfloor\frac{n}{k}\rfloor} \right) \bigoplus  \left( \bigoplus_{m} J_{0,\lceil\frac{n}{k}\rceil} \right).
$$
It follows from Lemma \ref{miller} that $A$ is conjugate to  $J_{0,n}^k$, so $A$ is an $k$th power. Now we observe that $J_{0,n} = A + \  \mathfrak J$, where $\mathfrak J$ is a junction matrix associated to the following partition of $n$:
$$\left( \underbrace{\lfloor\frac{n}{k}\rfloor, \ldots, \lfloor\frac{n}{k}\rfloor}_{k-m \trm{ terms }} ,
\underbrace{\lceil\frac{n}{k}\rceil, \ldots , \lceil\frac{n}{k}\rceil}_{m \trm{ terms }} \right).$$
Note that the parts of the partition $\lfloor\frac{n}{k}\rfloor$ and $\lceil\frac{n}{k}\rceil$ are at least $2$ since $n\geq 2k$, so by  Lemma~\ref{Lemma-junction-conjugate}, $\mathfrak J$ is an $k$th power too. Therefore $J_{0,n}$ is a sum of two $k$th powers.
\end{proof}

\section{Main Result}\label{main_result}

In this section we assemble results of
\cite{Sm} and \cite{Ki} and various results in the previous section to prove the main result of this article.

\begin{thm}
For all integers $k \geq 1$, there exists a constant $C_k$ depending only on $k$ such that for all $q > C_k$ and for all $n \geq 1$ every matrix in $M_n(\mathbb{F}_q)$ is a sum of two $k$th powers.
\end{thm}

\begin{proof}
We consider various cases. The symbol $p$ below denotes the characteristic of $\mathbb{F}_q$. \\

\noindent \textbf{(Case $k \geq 1$, $n =1$):} The result is due to Small \cite{Sm}. The constant $C_1$ obtained depends only on $k$, in fact it is $k^4$.\\

\noindent \textbf{(Case $k \geq 1$, $n =2$):} The result is due to the Kishore; see Theorem $1.1$ of \cite{Ki}. The constant $C_{2}$ in this case depends only on $k$. \\

\noindent \textbf{(Case $k \geq 1$, $n \geq 3$, $p = 2$):} The result is due to the Kishore;  see Theorem $1.2$ of \cite{Ki}  and the remark immediately after its proof [page 93. \cite{Ki}]. The constant $C_{3}$ depends only on $k$; indeed, the proof of Theorem $1.2$ immediately implies that the constant is in fact the one given by the more special case $n =2$ above.\\

\noindent \textbf{(Case $k \geq 1$, $n \geq 3$, $p \neq 2$):} By Proposition \ref{reduction} with the choice of constant $C = 0$, it suffices to restrict our attention  to matrices of the form $J_{\alpha,n}$ where $\alpha$ is a primitive element of $\mathbb{F}_q$, and also to the regular nilpotent square matrix of size $n$. By Proposition \ref{primitive_element}, which is valid for field of characteristic not $2$, it follows that there exists a constant $C_{4}$ that depends only on $k$ such that for all $q > C_{4}$ the matrix $J_{\alpha,n}$ is a sum of two $k$th powers. On the other hand, by Theorem \ref{theorem-power-nilpotent}, for every $n \geq 2k$, every $J_{0,n}$ is a sum of two $k$th powers, so there are only finitely many exceptions, namely those positive integers $n$ strictly less than $2k$. But, by Theorem \ref{C_depend_k_n} for each such $n$ there exists a respective constant $C(k,n)$ such that $J_{0,n}$ is a sum of two $k$th powers. Let $C_5$ be the maximum of the constants
$$
C_5 = \trm{max}(C_4, C(k,1), C(k,2), \ldots, C(k,2k-1)).
$$
Together with Theorem \ref{theorem-power-nilpotent}, it follows that  for all $q > \textrm{max}(C_5,2k)$, the Jordan nilpotent matrix $J_{0,n}$ is a sum of two $k$th powers as desired.

Let $C_6$ be the maximum of $C_1, C_2, C_3, C_5$ and $2k$; $C_6$ depends only of $k$. Then for $q > C_6$ and for all $n \geq 1$ every matrix is a sum of two $k$th powers.

\end{proof}

\section*{Acknowledgements}
\noindent It is a great pleasure to thank Michael Larsen for suggesting the question.
\bibliographystyle{elsarticle-num}

redpajama_set_name: RedPajamaArXiv


text:
Planinsko (în ) este un sat în comuna Banite, regiunea Smolean,  Bulgaria.

Demografie

La recensământul din 2011, populația satului Planinsko era de  locuitori. . Pentru % din locuitori nu este cunoscută apartenența etnică.

Note

Sate din regiunea Smolean

redpajama_set_name: RedPajamaWikipedia


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I lost my mother about a year ago to multiple health problems and had to make the worst and hardest decision of my life. This article will touch on how I've gotten through the grieving process and my life without my mom being in it.
Do you know someone who has recently lost a loved one? Here is some advice for what's comforting—and what's not.
When children experience grief during their early developmental stages, their emotional health is compromised. Our actions as adults can help them work through their grief and learn to trust again.
The final goodbye can be a daunting date, especially if it has been put off months from the death of your loved one. Know what to expect and how to make it through the day.
When my late husband lost his battle with cancer, I was fortunate enough to be present when he took his last breath. I learned what a privilege it was to be with a person at their life's end.
If you're like most people, you love your father and have a lifetime of cherished memories. This hub shares the author's intimate memories of her father's last years and what it meant to be a special part of his life. Be sure to keep your kleenex handy!
For me, the grief of losing my husband felt like the panic you feel just before you fall. This is my advice to people who are trying to comfort a friend or loved one who is grieving a loss.
Losing a parent to cancer is a deeply saddening experience. There are all kinds of tools for coping with anticipatory grief, but here are some tips I hope to share with you to keep you grounded as you go through this difficult time with your loved one.
Overtly religious condolences offered to non-believers during times of grief may feel fake, meaningless, or even rude. Learn how to deal with those feelings and understand what those sentiments mean.
My whole family came together to care for my terminally ill father. We actively chose to forge a new experience with him that was satisfying and fulfilling, without any lingering guilt or regret.


About 80% of Ugandans live in debt. This percentage is actually quite conservative with the figure closer to 90%. Financial indebt and poor management is rampant and has not spared the church. To put it plainly, most prayer requests in the church today are due to saints stuck in deep financial woes that can lead to serious consequences. Pastors and Ministers today find themselves in positions to give financial counseling. However due to lack of training sometimes this counselling fails to produce the required result and the sheep blame the shepards. In addition, the attitude towards money and wealth in the church needs to change. Most Christians do not have a healthy attitude or understanding of the financial Kingdom principles. Principles such as tithing, sowing and giving are most often misunderstood and at times may leave the Christian perplexed.
However, there are several financial kingdom principles found throughout the Bible and the purpose of the Market Place ministries is to search them out and provide a platform for the members of CCK to acquire knowledge about God's purposes concerning wealth, for without knowledge the people perish.
Scripture informs us that the LORD desires HIS chosen to make wealth to advance the Kingdom of the God and be a blessing to many. We have been told that we are the heads and not the tails, above and not beneath and we will lend but not borrow. Yet several even in the church struggle with debt and with daily financial battles in our businesses, homes and ministries. In order to succeed we need knowledge and application of godly financial principles.
The Market place ministries is taking a Biblical view on finances and wealth and going forth in the godly principles of wealth in the kingdom of the LORD. The Ministry plans to work at breaking the strongholds of debt and lack in CCK and impart godly financial and wealth building principles to the church.
1. God is the origin of all things including wealth.
6. Learn to be contentment with what you have but keep moving forward to what you want to achieve.
8. Don't' guarantee a loan of someone you don't know; and only guarantee what you are willing to lose.
To raise godly leaders in the church with a healthy view of financial kingdom principles (This is a working Mission that needs brain storming). The Market place ministry plans on building people who invest in others.
The CCK Market place leadership ministries is within the City Church of Kampala and is governed by the core values of CCK.


text:
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Renting video games online is gaining some serious popularity with hardcore and casual gamers alike. Whether you are just hearing about this industry for the first time or have been on the fence about joining here are 10 great reasons to start renting video games online.
Getting ganked by people who can only be decried as 100% Gankers! started to take over my entire league of legends game. The only good thing that has ever come from getting ganked was my change in attitude! I took it upon myself to man-up, get trained, and seek my sweet revenge on the ganker/s.
As a kid, I grew up playing video games – from Pong onward. I've always both enjoyed and been fascinated by them. I even did my dissertation research on the effects of video game violence on kids. So, I have a lot of knowledge about this topic given that I'm a life-long gamer, a psychologist who has researched their effects, and, now, a parent of two boys.
Looking at things from a different perspective, the research on neuroplasticity indicates that the brain really is like a muscle in that it can grow in response to stimuli. video games contain rich environments with many complex cognitive challenges that give brains quite a workout. There is a growing body of research to support that video game play improves cognitive functioning in many areas. Video or computer games are listed as one of 6 known "brain boosters" in the Feb/Mar 2009 issue of Scientific American Mind. Indeed, Microsoft is looking into the educational benefits of typical lol boost.
The lands of Valoran have always been affected by continuing conflicts and Rune wars till it came to an end by the super powers creating an Institute of War. It is created to settle disputes by league of legends that have their users with magic powers compete with each other as part of a team to rule the land of Valoran. You fight the battle with your trusted champions to win the game and rule the world of Valoran.
Gino Cappelletti – Cappelletti holds the record for the most points scored in a season and the most career point scored. Cappelletti scored 155 points in 1964 season, and was the AFL's most valuable player in 1964. During his career with the Boston Patriots, from 1960-1960, Cappelletti scored a total of 1,100 points.
When we talk about brainwashing, we think of military camps or cults. We can't help but think about physical and emotional torture. We imagine getting confined and detained against our own will. There is an element of helplessness when a person is being held captive against his own will and imposed upon certain values and beliefs in that current state.
In 2009 we wait and see the British Open and US Open to satisfy are golf taste. We will also be met by new NFL and college football seasons. With the World Series still out there after the conclusion of the World Baseball Classic we will have a new debate on the best baseball in the world. 2009 has many events past and to come that will be marked down forever.
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text:
package org.apache.commons.fileupload;

/**
 * This exception is thrown in case of an invalid file name.
 * A file name is invalid, if it contains a NUL character.
 * Attackers might use this to circumvent security checks:
 * For example, a malicious user might upload a file with the name
 * "foo.exe\0.png". This file name might pass security checks (i.e.
 * checks for the extension ".png"), while, depending on the underlying
 * C library, it might create a file named "foo.exe", as the NUL
 * character is the string terminator in C.
 *
 * @version $Id$
 */
public class InvalidFileNameException extends RuntimeException {

    /**
     * Serial version UID, being used, if the exception
     * is serialized.
     */
    private static final long serialVersionUID = 7922042602454350470L;

    /**
     * The file name causing the exception.
     */
    private final String name;

    /**
     * Creates a new instance.
     *
     * @param pName The file name causing the exception.
     * @param pMessage A human readable error message.
     */
    public InvalidFileNameException(String pName, String pMessage) {
        super(pMessage);
        name = pName;
    }

    /**
     * Returns the invalid file name.
     *
     * @return the invalid file name.
     */
    public String getName() {
        return name;
    }

}


The nominations are out for the 2019 Grammy Awards, including categories for jazz, soul, R&B, blues, American roots music and production. Among the big names this year are Tony Bennett and Diana Krall for their recent duets album 'Love is Here to Stay', they're up against Willie Nelson, Seal and Gregory Porter for his 'Nat King Cole and Me' project in the best vocal jazz album category.
Lalah Hathaway is nominated three times – for best R&B performance, best traditional R&B performance and best R&B album, while Jazz FM's own Marcus Miller is up for best contemporary instrumental album; he's up against Steve Gadd, Simon Phillips, Christian Scott aTunde Adjuah and Julian Lage.
Meanwhile, the hotley contested vocal jazz category contains Kurt Elling, Freddy Cole, Kate McGarry, Raul Midon and last year's winner Cecile McLorin Salvant. Arranger Vince Mendoza clocks up three nominations for his work with both Raul Midon and Gregory Porter, while Netflix's Quincy Jones documentary, produced by his daughter Rashida is up for best music film.


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text:
<?xml version="1.0" encoding="utf-8" ?>
<testSuite name="All-rows TestSuite" xmlns="http://NBi/TestSuite">
  <settings>
    <default apply-to="system-under-test">
      <connectionString>Data Source=.\SQL2017;Initial Catalog=AdventureWorks2017;User Id=sa;password=Password12!</connectionString>
    </default>
  </settings>
  <test name="With Predicate and with formula" uid="0000">
    <system-under-test>
      <execution>
        <query name="Select first departments">
          select row_count as DeptId, [Name], [GroupName] from [HumanResources].[Department] order by Name
        </query>
      </execution>
    </system-under-test>
    <assert>
      <all-rows>
        <variable column-index="0">DeptId</variable>
        <expression name="LogDepId">
          Log10(DepId)
        </expression>
        <predicate name="ModDepId">
          <more-than or-equal="true">10</more-than>
        </predicate>
      </all-rows>
    </assert>
  </test>
  <test name="With Predicate Null or empty" uid="0001">
    <system-under-test>
      <execution>
        <query name="Select first departments">
          select null as Name
        </query>
      </execution>
    </system-under-test>
    <assert>
      <all-rows>
        <alias column-index="0">Name</alias>
        <alias column-index="1">Name2</alias>
        <predicate name="Name" type="text">
          <empty or-null="true"/>
        </predicate>
      </all-rows>
    </assert>
  </test>
  <test name="With Predicate StartsWith" uid="0002">
    <system-under-test>
      <execution>
        <query name="Select first departments">
          select [Name] from [HumanResources].[Department] where Name like 'E%' order by Name
        </query>
      </execution>
    </system-under-test>
    <assert>
      <all-rows>
        <variable column-index="0">Name</variable>
        <predicate name="Name" type="text">
          <starts-with>E</starts-with>
        </predicate>
      </all-rows>
    </assert>
  </test>
  <test name="With Predicate EndsWith" uid="0003">
    <system-under-test>
      <execution>
        <query name="Select first departments">
          select [Name] from [HumanResources].[Department] where Name like '%e' order by Name
        </query>
      </execution>
    </system-under-test>
    <assert>
      <all-rows>
        <variable column-index="0">Name</variable>
        <predicate name="Name" type="text">
          <ends-with ignore-case="false">e</ends-with>
        </predicate>
      </all-rows>
    </assert>
  </test>
  <test name="With Predicate contains" uid="0004">
    <system-under-test>
      <execution>
        <query name="Select first departments">
          select [Name] from [HumanResources].[Department] where Name like '%en%' order by Name
        </query>
      </execution>
    </system-under-test>
    <assert>
      <all-rows>
        <variable column-index="0">Name</variable>
        <predicate name="Name" type="text">
          <contains ignore-case="true">en</contains>
        </predicate>
      </all-rows>
    </assert>
  </test>
  <test name="With Predicate is match regex" uid="0005">
    <system-under-test>
      <execution>
        <query name="Select first departments">
          select [Name] from [HumanResources].[Department]
        </query>
      </execution>
    </system-under-test>
    <assert>
      <all-rows>
        <variable column-index="0">Name</variable>
        <predicate name="Name" type="text">
          <matches-regex>^[A-z]+$</matches-regex>
        </predicate>
      </all-rows>
    </assert>
  </test>
  <test name="With Predicate is lower case" uid="0006">
    <system-under-test>
      <execution>
        <query name="Select first departments">
          select lower([Name]) from [HumanResources].[Department]
        </query>
      </execution>
    </system-under-test>
    <assert>
      <all-rows>
        <variable column-index="0">Name</variable>
        <predicate name="Name" type="text">
          <lower-case/>
        </predicate>
      </all-rows>
    </assert>
  </test>
  <test name="With Predicate is lower case" uid="0007">
    <system-under-test>
      <execution>
        <query name="Select first departments">
          select upper([Name]) from [HumanResources].[Department]
        </query>
      </execution>
    </system-under-test>
    <assert>
      <all-rows>
        <variable column-index="0">Name</variable>
        <predicate name="Name" type="text">
          <upper-case/>
        </predicate>
      </all-rows>
    </assert>
  </test>
  <test name="With Predicate is within-range" uid="0008">
    <system-under-test>
      <execution>
        <query name="Select first departments">
          select 10 union select 20 union select 30
        </query>
      </execution>
    </system-under-test>
    <assert>
      <all-rows>
        <variable column-index="0">Value</variable>
        <predicate name="Value" type="numeric">
          <within-range>[10;30]</within-range>
        </predicate>
      </all-rows>
    </assert>
  </test>
  <test name="With Predicate is within-list" uid="0009">
    <system-under-test>
      <execution>
        <query name="Select first departments">
          select 10 union select 20 union select 30
        </query>
      </execution>
    </system-under-test>
    <assert>
      <all-rows>
        <variable column-index="0">Value</variable>
        <predicate name="Value" type="numeric">
          <within-list>
            <item>10</item>
            <item>20</item>
            <item>30</item>
          </within-list>
        </predicate>
      </all-rows>
    </assert>
  </test>
  <test name="With Predicate is within-list" uid="0010">
    <system-under-test>
      <execution>
        <query name="Select first departments">
          select 10 union select 20 union select 30
        </query>
      </execution>
    </system-under-test>
    <assert>
      <all-rows>
        <variable column-index="0">Value</variable>
        <predicate name="Value" type="numeric">
          <any-of>
            <item>10</item>
            <item>20</item>
            <item>30</item>
          </any-of>
        </predicate>
      </all-rows>
    </assert>
  </test>
  <test name="With multiple expressions" uid="0011">
    <system-under-test>
      <execution>
        <query name="Select first departments">
          select 10 union select 20 union select 30
        </query>
      </execution>
    </system-under-test>
    <assert>
      <all-rows>
        <expression name="div">[#0] / 2</expression>
        <expression name="multi">[#0] * 2</expression>
      </all-rows>
    </assert>
  </test>
  <test name="With expression written with a native script" uid="0030">
    <system-under-test>
      <execution>
        <query name="Select first departments">
          select row_count as DeptId, [Name], [GroupName] from [HumanResources].[Department] order by Name
        </query>
      </execution>
    </system-under-test>
    <assert>
      <all-rows>
        <variable column-index="0">DeptId</variable>
        <expression name="IntegerDepId">
          <script language="native">
            DeptId | numeric-to-integer
          </script>
        </expression>
        <predicate name="IntegerDepId">
          <more-than or-equal="true">10</more-than>
        </predicate>
      </all-rows>
    </assert>
  </test>
</testSuite>