There's a lot of rumours and not a lot of information going around; so, is Curtin really introducing trimesters from 2020?
At the moment, Curtin looks set to introduce a third study period into the 2020 academic calendar. But that doesn't make it a trimester, since it won't be compulsory for all students.
Deputy Vice Chancellor Academic, Professor Jill Downie, who is leading the push for the proposal, said the changes are focused on making things better for students.
"It really is about flexibility for students," she said.
"There are a considerable amount of students who want to fast-track, and there are a considerable amount who would like an extended period over the summer for paid work.
Curtin Student Guild President, Liam O'Neill, said despite what the University says, the proposal is for a trimester system.
"A trimester is an academic year divided into three periods. That is what Curtin is proposing," he said.
National Tertiary Education Union Curtin branch president Tony Snow said offering three study periods every year could mean courses run when they aren't viable.
"If you follow this road, sooner or later you might get to the stage where you say, 'well, sorry, but it's going to have to be three continuous semesters and that's the study pattern for everybody,'" he said.
Mr O'Neill said no matter what form the changes take, they'll only end up hurting students.
"This model undermines the standard of our education, lowering the quality of our degrees and damaging graduate outcomes," he said.
On the topic of mental health, Mr O'Neill said the consequences of the changes could be "disastrous".
"With Universities Australia research indicating that two-thirds of students are living below the poverty line, and headspace research indicating huge rates of mental health issues among students, all that losing study weeks and tuition free weeks does is make this much worse," he said.
Professor Downie, whose portfolio includes student support services, said she believed the changes would have the opposite effect.
"We don't think flexibility and choice is something that will increase mental health issues," she said.
The current debate over the academic calendar began in 2016, when a working party began considering changes to Curtin's academic calendar at the request of the Academic Board.
At the conclusion of that process in September 2017, the Academic Board accepted the party's recommendation that the current calendar be maintained.
Guild breathed a sigh of relief, having avoided the possibility of losing a tuition free week under one of the suggested proposals.
But, less than a year after the Academic Board's decision, the University approached the Guild to discuss its intention to introduce a third study period over summer—available to all students.
Mr O'Neill said Curtin's backflip sets a dangerous precedent moving forward.
"This is disappointing in the extreme," he said.
"How can the University now expect anyone to engage in a consultative process if we know that they will never honour it?
"While the University claims it will be optional, it will be very easy to change this down the track, and I think we all know better than to believe what Curtin says," he said.
Mr Snow said the University's academic staff were just as upset as students.
"I think this is really what has angered most of our staff at the University, [it's] that we went through this procedure last year, exhaustively," he said.
Mr Snow said the proposal is a new one, with a trimester-free academic calendar already set last year.
"In November, we actually sat down with UAB and they came back with a revised 2020 calendar, which included six week intensive periods," he said.
Professor Downie said the Academic Board's decision had been made under different circumstances, before Curtin's Mauritius and Dubai campuses had been opened.
"Things have changed, as you would expect in an agile and flexible university," she said.
"The proposed changes are also designed to establish a common calendar across campuses in WA, Singapore, Mauritius and Malaysia.
Mr O'Neill rejected that suggestion, and said it was a weak justification by Curtin.
"I think that this is a vastly overstated benefit that does not stack up," he said.
"This change only moves Bentley, Kalgoorlie, Malaysia and Mauritius in line with Curtin Singapore and the City [Business and Law campuses].
According to a calendar published by Guild, students' summer breaks could be up to five weeks longer under the proposed changes.
Professor Downie said this would give students the chance to better position themselves to enter the workforce.
"For those students not wishing to study across the summer period, the proposed changes will provide enhanced opportunities to engage in paid work, placements or volunteer work," she said.
But Mr O'Neill said the changes will result in the complete opposite.
"The issue is that students will have to fit in the same amount of study, placements and other commitments within a much more rigid timeframe," he said.
"Flexibility will be reduced and those who work part-time throughout the year will be at a disadvantage.
"If the trimester model becomes compulsory, and I believe it is only a matter of time before this occurs, these issues will become even more pronounced.
"Employers have consistently pointed to the need for students to do more than just study for their degree at University.
Journalism and Law student Eliza Lyon said the changes would hit students who aren't originally from Perth hard.
"I think trimesters could be quite detrimental to Curtin," she said.
"The fact it will cut down the Christmas break will only frustrate students who either go away in the break or use the time with family and friends.
For other students, like International Relations and Economics student Luke Jakovcevic, it's the lack of clear information that worries him the most.
"The main thing I'm worried about is that they haven't released much of their plans, just that they're planning on introducing three study periods," he said.
"They haven't told us how it's really going to impact us, so we're all just guessing at the moment.
Staff oppose the change too?
Mr Snow said staff are worried the changes will make it even more difficult to give students the support they need.
"I think most academic staff would say from a pedagogical point of view, it's not good practice," he said.
"From a point of view of being prepared for the lectures and keeping up the workload, including the marking, feedback to students, all those sorts of things, it's just simply making that a lot more difficult.
Curtin Law School Lecturer, Dr Hugh Finn, said the proposal would mean a huge workload for students and staff who are part of courses already delivered in trimesters.
"The implications of the proposed academic calendar change are significant for Curtin Law School students because the teaching model for the LLB degree is based on a trimester system for the second- and third-year units," he said.
"If the existing trimester system for the law degree were simply to convert to the proposed three study period calendar, then law students would be in class or taking exams for 42 weeks a year.
However, Dr Finn could also see some upsides to the proposed changes.
"The main benefit [for law students] is that each study period would have 12 teaching weeks, rather than the 10 teaching weeks in the current trimester calendar," he said.
No study week the biggest loss?
Many students have pointed to the removal of the study week before exams as the worst part of the changes, including Business Information Systems student, Tom Brewster.
"That week gives you a mental break and allows you to get all of your notes and thoughts together," he said.
That sentiment was echoed by Mr Snow, who said the final weeks of each study period would only become more stressful.
"How are they going to be able to turn that around and actually get it back to students so the student can actually get feedback, can understand why he may have done poorly in that or maybe done well in it, and prepare for an exam at the same time?" he said.
Mr Snow also said he could see students doing poorly in exams because of the rushed nature of the proposed system.
"If you've got exams in that first week and you're still having examinable material in the last week of classes, it does make it very difficult because that turn-around where you're trying to develop the understanding of the work, trying to get to a point where you actually remember it and understand it, takes a bit of time and it's just not available," he said.
Professor Downie said while no decisions had been made yet, the University was working on ways to "streamline" the examination process, including overhauling the Board of Examiners system, and holding exams at outside venues.
Professor Downie said students still have time to make their voices heard.
"At the request of the Curtin Student Guild, the University has agreed to extend the consultation period to provide more opportunities for students to engage with the proposal and provide feedback," she said.
Mr O'Neill said the Guild is currently planning their next steps and encouraged students who are against the proposed changes to sign their petition.
In a statement released on June 18, Professor Downie said the University is happy to meet with students, and will release information about a consultation process shortly.
Professor Downie's office can be contacted here.


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text:
H.R. 3207 (116th): To designate the facility of the United States Postal Service located at 114 Mill Street in Hookstown, Pennsylvania, as the "Staff Sergeant Dylan Elchin Post Office Building".
To designate the facility of the United States Postal Service located at 114 Mill Street in Hookstown, Pennslyvania, as the "Staff Sergeant Dylan Elchin Post Office Building".
Sponsor. Representative for Pennsylvania's 17th congressional district. Democrat.
Enacted — Signed by the President on Dec 3, 2020
This bill was enacted after being signed by the President on December 3, 2020.
17 Cosponsors (9 Republicans, 8 Democrats)
What legislators are saying
"house passes bill to designate hookstown post office as the staff sergeant dylan elchin post office building"
— Rep. Conor Lamb [D-PA17, 2019-2022] (Sponsor) on Oct 16, 2019
"the daily leader: wednesday, october 16, 2019"
— Rep. Steny Hoyer [D-MD5] on Oct 15, 2019
"the weekly leader: friday, october 11, 2019"
More statements at ProPublica Represent...
Considered by House Committee on Oversight and Reform
A committee held a hearing or business meeting about the bill.
Considered by Senate Committee on Homeland Security and Governmental Affairs
Updated bill text was published as of Reported by Senate Committee.
The bill was passed by both chambers in identical form. It goes to the President next who may sign or veto the bill. The vote was by Unanimous Consent so no record of individual votes was made.
This bill was introduced in the 116th Congress, which met from Jan 3, 2019 to Jan 3, 2021. Legislation not passed by the end of a Congress is cleared from the books.
GovTrack.us. (2023). H.R. 3207 — 116th Congress: To designate the facility of the United States Postal Service located at 114 Mill Street …. Retrieved from https://www.govtrack.us/congress/bills/116/hr3207
"H.R. 3207 — 116th Congress: To designate the facility of the United States Postal Service located at 114 Mill Street …." www.GovTrack.us. 2019. January 29, 2023 <https://www.govtrack.us/congress/bills/116/hr3207>
To designate the facility of the United States Postal Service located at 114 Mill Street in Hookstown, Pennsylvania, as the "Staff Sergeant Dylan Elchin Post Office Building", Pub. L. No. 116-197, H.R. 3207, 116th Cong. (2020).
|date=June 11, 2019
|quote=To designate the facility of the United States Postal Service located at 114 Mill Street …

redpajama_set_name: RedPajamaCommonCrawl


I remember reading a book about Disciplines. It may have been entitled as the "Disciplines of Discipleship", but I honestly forget. (My apologies to the author!) But I learned there was a need to organize my life with some good routines that would help me in my daily growth and allow me to be more effective in my ministry.
Now, we know that we must always rely on the Spirit of God to bless our work, but even He wants us to practice good habits in our lives. Paul said in 1 Corinthians 9:27 ESV, "But I discipline my body and keep it under control, lest after preaching to others I myself should be disqualified."
First, we build up our strength and endurance to be ready when trials and difficulties come along. We often think of "boot camp" for a young soldier. There he or she is expected to build up their bodies in strength and to practice hard at their routines so they will be ready to face an enemy. We see this also in sports when training camp does the same thing. I've often admired the good soccer players who just keep running almost the whole game! I am impressed with their endurance! But I realize that couldn't have happened without days and weeks of constantly running to build up their stamina.
Second, practicing good discipline helps us "...to be ready for every good work" Titus 3:1. Knowing the Word of God, enjoying intimate moment-by-moment fellowship with your Savior, keeping your heart clean and daily confessing your sin and confessing your need of God's guidance permits you to "be ready" for all the opportunities the Lord provides for you to serve Him. Paul said, "For bodily exercise profits a little, but godliness is profitable for all things..." 1 Timothy 4:8.
Third, we not only add to our lives good habits, but we discipline ourselves to avoid the things that can harm us. 2 Timothy 2:4 ESV says, "No soldier gets entangled in civilian pursuits, since his aim is to please the one who enlisted him." This is also reinforced when someone came to Jesus offering to "follow Him" but then spoke of other priorities. Luke 9:57-62 speak of three would-be disciples that didn't have the disciplines they needed to truly follow the Lord. They were preoccupied with other "priorities."
Now, discipline by itself is not necessarily that great. We can actually make more of discipline than our relationship with Christ. The point of discipline is not the discipline itself. It's to help us be used of our Savior effectively for His glory. But without a good relationship with Christ all our ministry is "service without love." And that's why the Lord was upset with the Ephesians in Revelation 2:4. They were more in love with their ministry than they were with Christ.
Keep your love intimate with Jesus and practice good disciplines that will help you to "serve Him in love."


text:
#region License

#endregion

using System;
using System.IO;
using System.Net;
using System.Reflection;
using System.Text;
using System.Threading;

namespace CG.Web.MegaApiClient
{
    public class WebClient : IWebClient
    {
        private const int DefaultResponseTimeout = Timeout.Infinite;
        private const uint BufferSize = 8192;

        private readonly int _responseTimeout;
        private readonly string _userAgent;

        public WebClient()
            : this(DefaultResponseTimeout)
        {
        }

        internal WebClient(int responseTimeout)
        {
            this._responseTimeout = responseTimeout;
            this._userAgent = this.GenerateUserAgent();
        }

        public string PostRequestJson(Uri url, string jsonData)
        {
            using (MemoryStream jsonStream = new MemoryStream(jsonData.ToBytes()))
            {
                return this.PostRequest(url, jsonStream, "application/json");
            }
        }

        public string PostRequestRaw(Uri url, Stream dataStream)
        {
            return this.PostRequest(url, dataStream, "application/octet-stream");
        }

        public Stream GetRequestRaw(Uri url)
        {
            HttpWebRequest request = this.CreateRequest(url);
            request.Method = "GET";

            return request.GetResponse().GetResponseStream();
        }

        private string PostRequest(Uri url, Stream dataStream, string contentType)
        {
            HttpWebRequest request = this.CreateRequest(url);
            request.ContentLength = dataStream.Length;
            request.Method = "POST";
            request.ContentType = contentType;

            using (Stream requestStream = request.GetRequestStream())
            {
                dataStream.Position = 0;

                int length = (int)Math.Min(BufferSize, dataStream.Length);
                byte[] buffer = new byte[length];
                int bytesRead;

                while ((bytesRead = dataStream.Read(buffer, 0, length)) > 0)
                {
                    requestStream.Write(buffer, 0, bytesRead);
                }
            }

            using (HttpWebResponse response = (HttpWebResponse)request.GetResponse())
            {
                using (Stream responseStream = response.GetResponseStream())
                {
                    using (StreamReader streamReader = new StreamReader(responseStream, Encoding.UTF8))
                    {
                        return streamReader.ReadToEnd();
                    }
                }
            }
        }

        private HttpWebRequest CreateRequest(Uri url)
        {
            HttpWebRequest request = (HttpWebRequest)WebRequest.Create(url);
            request.Timeout = this._responseTimeout;
            request.UserAgent = this._userAgent;

            return request;
        }

        private string GenerateUserAgent()
        {
            AssemblyName assemblyName = Assembly.GetExecutingAssembly().GetName();
            return string.Format("{0} v{1}", assemblyName.Name, assemblyName.Version.ToString(2));
        }
    }
}

redpajama_set_name: RedPajamaGithub


Coole Cleaners was started by Christopher Stuckey in 2007 in Wilmington, NC. Since then we have made it our priority to help homeowners everywhere in Wilmington manage their housework and balance their lives and budget. We strive in providing a more personal maid service at a professional level, but yet at an affordable cost. Coole Cleaners is fully licensed and insured. Our employees go through a background check, drug screening, and are covered by workers compensation. Coole Cleaners tries to eliminate all your worries and gain your trust in having a cleaning service in your home.
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text:
Business Beyond COVID‑19
Author: Michael Marais
Post author By Michael Marais
On January 25th join business and legal leaders from a real rainbow of united nations, including Australia, Brazil, India, Japan, New Zealand, South Africa, the UK and USA, to conclude the 2020-21 Business Beyond Covid-19 on-line event series.
Over the past year, The Legal 500 joined with The Blended Capital Group to host a dynamic debate and exploration of how business in different countries across the world has mitigated the seemingly never-ending uncertainty and risks of the Global Pandemic. Now, in our closing session before offering recommendations to the UN World Investment Forum we are exploring how business worldwide is preparing for a vaccine-driven return to a new form of reality, hopefully, during an improving 2021.
Leading legal minds:
The on-line event series was kicked off by Judge Professor Mervyn King of South Africa and Professor Paul Q. Watchman, from the UK where he was a former Senior Partner at Freshfields Bruckhaus Deringer. Both Professors have kindly supported the on-line series from the start and will help deliver final recommendations to a forthcoming on-line event supporting the UN global investment event. You will be advised of the UN event date/time shortly.
REGISTER: Double session choice for Time Zones:
Respecting time zones ranging from Tokyo to San Francisco we will convene the event twice during the day at 0800hrs UK and then again at 1500hrs UK. Register here, selecting the option most suitable for your time zone.
We would like to acknowledge our generous supporters throughout the global series including:
United Nations South Africa;
Oriental Consultants Global Co., Ltd. (Japan);
Mwezi Co., Ltd. (Kenya);
Future-Fit Foundation (UK Charity);
The Tony Elumelu Foundation (Nigeria); and
Business Unity South Africa (BUSA SA).
Australia Forum
Australia locked down early, created a widely heralded National Cabinet to forge a cohesive response to Covid-19 and managed to limit deaths from the virus to 128 by late July. However, a surge in infections saw the State of Victoria reimpose a lockdown on July 9 attracting global media attention and by early August the State recorded its' peak number of daily infections at 727[1]. As of August 17, Australia had reported 23,599 confirmed cases and 421 deaths. Australia's call during the World Health Assembly[2] for an "impartial, independent and comprehensive investigation into the handling of the Covid-19 outbreak" sparked a furious diplomatic response from China that looks set to threaten the two country's critical trading relations.
The second event in the global series took place on August 27th 2020. You can find out more about the event here or watch the recording of the discussion below.
[1] https://theconversation.com/two-weeks-of-mandatory-masks-but-a-record-725-new-cases-why-are-melbournes-covid-19-numbers-so-stubbornly-high-143898
[2] https://www.who.int/about/governance/world-health-assembly
© 2021	Business Beyond COVID‑19


text:
\section{Introduction}

Planetary Nebulae (PNe) are the result of the evolution of low- and
intermediate-mass stars (1$-$8\,M$_{\odot}$), where the circumstellar gas
previously expelled during the Asymptotic Giant Branch (AGB) phase is ionized
by the central star (CS). The infrared emission from dusty circumstellar shell
that developes at the end of the AGB can remain strong during the PN phase, and thus
be easily detectable with infrared telescopes.

An interesting class of PNe are those with C-rich Wolf-Rayet ([WC]-type) central
stars (the so-called `WCPNe'), which have little or no H in their atmospheres.
Observations completed by the Infrared Space Observatory (ISO) suggest
that the shells around some WCPNe exhibit a characteristic dual-dust  (or
mixed) chemistry \citep{Waters1998a, Waters1998b, Cohen1999, Cohen2002}, which
is not yet well understood. This is deduced from the simultaneous  presence of
the well-known family of infrared bands at 3.3, 6.2,
``7.7'', 8.6, and 11.3\,$\mu$m due to Policyclic Aromatic Hydrocarbons (PAHs: C-rich)
\citep{Leger1984, Allamandola1985}, and several relatively broad  bands (the
strongest at 23.5, 27.5, and 33.8\,$\mu$m) attributed to crystalline  silicates
(O-rich), such as olivines and pyroxenes \citep{Waters1998a} in the  ISO
spectra. The mixed chemistry phenomenon appears to be common among  late-type/cool
WCPNe, while only one (NGC 5315) out of 6 early-type/hot WCPNe  observed with
ISO shows this dual-dust chemistry. In total  about 50\% of the WCPNe show
mixed chemistry \citep{Cohen2002,  DeMarco-Soker2002}. Note, however, that PAH
features seem to be present in  most of the 16 WCPNe analyzed by
\citet{Szczerba2001}, with the clear exception of K\,2-16. Therefore, we should
keep in mind that, due to the poor quality of ISO band 4
data, the true percentage of WCPNe with mixed chemistry may be underestimated.

In this {\it Letter}, we report the results of a systematic,
spectroscopic analysis completed with the Infrared Spectrograph (IRS, \citealt{Houck2004})
onboard the Spitzer Space Telescope \citep{Werner2004} of a sample of 26
Galactic Bulge (GB) PNe around central stars, which are classified as
late-type/cool [WC], early-type/hot [WC], weak emission-line stars ({\it
wels}) \citep{Tylenda1993}, and neither [WC] nor {\it
wels}. We also briefly discuss the remaining 14 non-GB
PNe in our sample. Section 2 describes the infrared observations carried out and the data reduction
performed, while the results obtained are presented in Sect. 3. Finally, in
Sect.4 we discuss our findings in the context of stellar evolution in the
Galactic Bulge.

\begin{table*}
\caption{The sample of 21 Galactic Bulge PNe with dual-dust chemistry observed
 with Spitzer/IRS (program $\#$3633).}
\label{GBdualdust}
\begin{tabular}{r l c c c c c c}
\hline
\hline
IRAS name    & PN name  & R.A.       & DEC       &  Spectral type$^{\mathrm{a}}$         & Optical diameter$^{\mathrm{d}}$ & distance from GC & dust type \\
             &          & J2000.0    & J2000.0   &                        & (arcsec)                        & ($\degr$)        &           \\
\hline
17480$-$3023 & M 3-44   & 17 51 18.7 & -30 23 53 & [WC11]$^{\mathrm{b}}$  & 4.4 & 1.91 & mixed \\
17549$-$3347 & H 1-43   & 17 58 14.5 & -33 47 37 & [WC11]$^{\mathrm{b}}$  & 2.0 & 5.56 & mixed \\
17434$-$3307 & M 1-27   & 17 46 45.5 & -33 08 35 & [WC11]$^{\mathrm{b}}$ & 5.3 & 4.21 & mixed \\
18061$-$2505 & MaC 1-10 & 18 09 12.4 & -25 04 35 & [WC8]$^{\mathrm{b}}$   &10.0 & 6.52 & mixed \\
17355$-$2206 & M 1-25   & 17 38 30.3 & -22 08 39 & [WC6]$^{\mathrm{b}}$   & 4.6 & 6.99 & mixed \\
17425$-$2056 & M 3-15   & 17 45 31.8 & -20 58 06 & [WC5]$^{\mathrm{b}}$   & 4.2 & 7.97 & mixed \\
17388$-$2440 & Hb 4     & 17 41 53.1 & -24 42 03 & [WC4]$^{\mathrm{b}}$   & 6.2 & 4.32 & mixed \\
18259$-$3132 & Cn 1-5   & 18 29 11.7 & -31 29 59 & [WC4]$^{\mathrm{b}}$   & 7.0 & 9.75 & mixed \\
17389$-$2409 & M 2-14   & 17 41 57.7 & -24 11 11 & {\it wels}$^{\mathrm{b}}$    & 0.0 & 4.82 & mixed \\
17496$-$2221 & M 1-31   & 17 52 41.4 & -22 21 57 & {\it wels}$^{\mathrm{b}}$    & 0.0 & 6.76 & mixed \\
18006$-$3117 & M 2-27   & 18 03 52.6 & -31 17 47 & {\it wels}$^{\mathrm{b}}$    & 4.8 & 4.60 & mixed \\
18054$-$2217 & M 1-40   & 18 08 26.0 & -22 16 53 & {\it wels}$^{\mathrm{b}}$    & 5.0 & 8.40 & mixed \\
18094$-$2450 & H 1-61   & 18 12 34.0 & -24 50 01 & {\it wels}$^{\mathrm{b}}$    & 0.0 & 7.27 & mixed \\
17156$-$3135 & Th 3-4   & 17 18 51.9 & -31 39 06 & none$^{\mathrm{b}}$    & 0.0 & 6.38 & mixed \\
17178$-$2900 & M 3-38   & 17 21 04.3 & -29 02 60 & none$^{\mathrm{c}}$    & 0.0 & 5.37 & mixed \\
17217$-$2803 & M 3-8    & 17 24 52.2 & -28 05 54 & none$^{\mathrm{b}}$    & 5.4 & 4.64 & mixed$^{\mathrm{e}}$ \\
17262$-$2623 & H 1-16   & 17 29 23.4 & -26 26 05 & none$^{\mathrm{b}}$    & $<$5.0 & 4.37 & mixed \\
17521$-$2144 & Hb 6     & 17 55 07.1 & -21 44 41 & none$^{\mathrm{b}}$    & 5.0 & 7.50 & mixed \\
17523$-$3033 & H 1-40   & 17 55 36.0 & -30 33 32 & none$^{\mathrm{b}}$    & 3.8 & 2.71 & mixed \\
18006$-$3241 & H 1-50   & 18 03 53.5 & -32 41 42 & none$^{\mathrm{b}}$    & $<$10. & 5.43 & mixed \\
18100$-$3220 & H 1-62   & 18 13 17.9 & -32 19 42 & unknown                & 0.0 & 6.85 &  mixed \\
\hline
\hline
\end{tabular}
\smallskip
\smallskip

\begin{tabular}{p{\textwidth}}
$^{\mathrm{a}}$ none - neither [WC]~nor {\it wels};
$^{\mathrm{b}}$ \cite{Gorny2004};
$^{\mathrm{c}}$ \cite{Gorny2008};
$^{\mathrm{d}}$ Strasbourg-ESO Catalogue of Galactic Planetary Nebulae, \\
\cite{Acker1992}, 0.0 - PN is a point-like
source; $^{\mathrm{e}}$ PAHs presence is tentative.
\end{tabular}
\end{table*}

\section{Observations and data reduction}

The spectral data were taken with Spitzer/IRS under a General Observer
program ($\#$3633, PI, M. Bobrowsky). We obtained Spitzer/IRS spectra of a
sample of 40 PNe\footnote{Plots of all spectra are available online. Table\,1
and online Tables\,A.1 and A.2 collect relevant information for our sample of
PNe.} in the 5.2$-$37.2 $\mu$m range by using the Short-Low (SL: 5.2$-$14.5
$\mu$m; 64$<$R$<$128), Short-High (SH: 9.9$-$19.6 $\mu$m; R$\sim$600) and
Long-High (LH: 18.7$-$37.2 $\mu$m; R$\sim$600) modules. All sources in our
sample are bright at mid-infrared wavelengths, and it was easy to achieve a
signal-to-noise ratio higher than 50 in each of the three modules SL, SH, and LH by using two
exposure cycles of 6 s. The starting point for our interactive data reduction
were the coadded 2D flat-fielded images (one for each nod position; pipeline
version 12.3), in which ``rogue'' pixels were cleaned.
For SL, the two nod positions were subtracted in order
to remove the sky background. However,
for the high-resolution modules no background subtraction was done since no sky
measurements were taken and the SH and LH slits were too small for on-slit
background subtraction. The spectra for each nod position were extracted from
the 2D images, and wavelength and flux calibrated using the Spitzer IRS Custom
Extractor (SPICE) with a point-source aperture. The 1D spectra were cleaned for
bad data points, spurious jumps and glitches, combined and merged into one
single spectrum per module for each target using SMART\footnote{SMART was
developed by the IRS Team at Cornell University and is available through the
Spitzer Science Center at Caltech.} \citep{Higdon2004}.
Finally, when required we scaled the LH spectra to the available Midcourse Space Experiment
\citep[MSX;][]{Egan_etal_2003} photometry at
21.34\,$\mu$m, and the corresponding SL and SH spectra were also scaled upward to match the LH spectra.
This is because some of the PNe studied here are extended sources (optical
diameter $>$4"), giving a flux loss at the smaller SL and SH apertures. The
MSX photometry at 21.34 $\mu$m was selected as a reference point for flux
calibration because this filter is less affected by
nebular emission lines and dust emission features.

\vspace{-0.09in}
\section{Results}
In Table\,1, we present the names (columns 1 \& 2), coordinates (columns 3 \& 4),
spectral type (column 5), optical diameter (column 6), distance from Galactic
Center (GC: column 7), and dust type (column 8) for the 21 PNe in our
sample, which belong to the Galactic Bulge\footnote{A PN was counted as a member
of the GB when its angular distance from the GC was $<$\,10$\degr$ and its
optical angular diameter was $<$\,10$\arcsec$.} and show dual-dust
chemistry. The total number of GB PNe observed by us is 26, so the
occurrence of dual-dust chemistry (21/26) is much higher than the 6/11
observed in the sample of GB PNe discussed by \citet{Gutenkunst2008}.
   \begin{figure}
   \centering
   \includegraphics[bb=0 22 435 770,width=8cm,clip]{1457Fig1.eps}
   \caption{Spitzer/IRS spectra of four representative GB PNe from our sample
(from top to bottom: late-type/cool [WC], early-type/hot [WC], {\it wels} and
neither [WC] nor {\it wels}). Note the simultaneous detection of PAH emission
features in the 6$-$14 $\mu$m range (see Fig.\,2) together with prominent
crystalline silicate emission features (marked with dotted vertical lines at
23.5, 27.5 and 33.8\,$\mu$m) beyond 20 $\mu$m, in all cases.}
   \label{GB4whole}
   \end{figure}
   \begin{figure}
   \centering
   \includegraphics[bb=0 22 435 770,width=8cm,clip]{1457Fig2.eps}
   \caption{Same spectra as in Fig.\,1, but for wavelenghts $<$ 14\,$\mu$m.
The positions of typical PAH features at 6.2, ``7.7'', 8.6 and 11.3 $\mu$m are
indicated by dotted vertical lines.}
   \label{GB4zoom}
   \end{figure}
Figure\,1 shows representative Spitzer/IRS spectra corresponding to 4 classes of
PN CSs from Table\,1: late-type/cool [WC] (H 1-43); early-type/hot [WC] (Cn
1-5); {\it wels} (M 1-40); and neither [WC] nor {\it wels} (H 1-16). In Appendix
B, we present spectra for all 21 PNe from Table\,1 (see Figs.\,B1, B2 and B3),
and for the remaining 5 GB PNe in our sample (Table\,A1 in Appendix A
contains relevant information for these sources), which show only the
presence of silicates (Fig.\,B4). The spectral type of the CS is shown below the
PN name and  ``none'' is used instead of ``neither [WC] nor {\it wels}''. All
spectra exhibit a strong infrared continuum emission, which peaks  between 20 and
40\,$\mu$m, as observed in many other PNe detected by ISO. This strong
infrared excess is produced by the thermal emission of dust still present in
their remnant circumstellar AGB shells. Overimposed on this infrared continuum
we can clearly distinguish the presence of narrow nebular emission lines.

In the spectra, we can also see broad solid-state features from the dust
grains. The simultaneous presence of PAH emission features (C-rich) in the
6$-$14\,$\mu$m range (Fig.\,2 and the right panels of
Figs.\,B1, B2, and B3) and prominent crystalline silicate emission features (O-rich) beyond 20 $\mu$m
(Fig.\,1 and the left panels of Figs.\,B1, B2 and B3)
illustrates the dual-dust chemistry in all 21 GB PNe. The mixed chemistry is seen not only in
late-type, but also in all observed early\footnote{Note that Cn 1-5 was
previously observed with ISO, although the spectrum was of bad quality and the
presence of dual-dust chemistry was uncertain in this early-type
WCPN.}-type WCPNe. However, the most striking result is that dual-dust chemistry
was detected in {\it all} observed {\it wels} as well as in some other GB
PNe with CSs, which are very likely of none of the above types as
inferred from their optical  spectra (see Table\,1). Spectral type of H 1-62 is unknown.
The class of GB PNe with neither [WC] nor
{\it wels} CSs is investigated in a subsequent paper
(G\'orny et al.~2009, in preparation).

A broad, dust feature around 10\,$\mu$m is seen in a few PNe. This feature
appears to be due to amorphous silicates and is seen only in 4 (Th 3-4, M 3-38, M
3-8 and H 1-40 - see Figs.\,B2 and B3) out of the 21 GB PNe with dual-dust
chemistry. The only other known case of WCPN with such feature is SwSt\,1
\citep[see e.g.][]{Szczerba2001}. By interpreting the strength of the 11.3\,$\mu$m PAH
band as a measure of the PAH abundance, we see that these 4 sources (except M
3-38) have the lowest PAH content (detection of PAHs is even tentative in
the case of M 3-8 - see Table\,1 and Fig\,B3). Surprisingly, this 10\,$\mu$m
band is seen in most non dual-dust chemistry GB sources in our
sample, except H 1-12, for which we have only SH observations and we cannot
deduce the presence of this band, and M 3-13 whose
spectrum is of rather low quality (see Fig.\,B.4). These sources show only weak crystalline silicate
features. The sources with 10\,$\mu$m dust feature are
discussed in more detail by G\'orny et al.\,(2009, in preparation).

There are 14 additional PNe in our sample, which are not counted as GB sources.
The relevant information for these sources is available only online and is
collected in Table\,A2 with spectra shown in Figs.\,B5
and B6. Among them, there are 6 PNe with dual-dust chemistry (3 early-[WC]; 2 {\it wels} and one with
unknown type CS). Surprisingly, there is one {\it wels}\footnote{The three
{\it wels} PNe are located only slightly further than 10$\degr$ from the
Galactic Centre, so they may as well belong to the GB population.} (M\,1-20)
that does not display the signs of O-rich dust, but only PAHs and a broad plateau emission
between 11 and 14\,$\mu$m (see Fig.\,B5), attributed to primordial C-rich
material, also observed in proto-PNe \citep[e.g.][]{Kwok2001}. This PN,
and another C-rich source (Tc 1) showing unusual, hydrogenated PAHs (see Fig.\,B6)
will be discussed in more detail in a forthcoming paper.
We note the exceptional source GLMP\,698 (with CS
classified as [WC8]), for which only weak bands of crystalline silicates are
detected (similar to ISO target K\,2-16 with a CS classified as [WC\,11]).
Another 5 sources show only silicates, with no indication of C-rich dust.
Out of the 14 non-GB PNe, there are 5 sources showing both, amorphous and
crystalline silicates.

\section{Discussion}

The origin of the dual-dust chemistry in PNe is still unclear. In general,
several mechanisms may play a role:

{\bf i}) a  very late thermal pulse at the end of the AGB phase (or  just
after), which may turn an O-rich outflow into a C-rich one \citep{Waters1998a}.
Due to the enhanced mass loss, high temperature annealing may lead to
the crystallization of amorphous silicates  \citep[e.g.][]{Sylvester1999}, while
PAHs form from the newly released C-rich gas;

{\bf ii}) evaporation of Oort-like cloud comets may release  crystalline
silicates \citep{Cohen1999}, while PAHs form after transformation of
the star into C-rich one;

{\bf iii}) the central star of the PN may exist in a binary system  with a
circumbinary O-rich disk that was formed long before the PN. Low temperature
crystallization may take place in such a disk
\citep{Molster1999}. PAHs would form after transformation of the O-rich star
into a C-rich one;

{\bf iv}) a low-mass main sequence star, brown dwarf, or
planet may spiral into the
AGB star, inducing a chemistry change as a consequence of extra-mixing
\citep{DeMarco-Soker2002}. High temperature annealing of amorphous silicates
and PAH formation from the newly released C-rich gas could be invoked;

{\bf v}) for intermediate-mass stars, the strong mass loss may induce a
sudden  deactivation of the {\it hot bottom burning}  \citep[see
e.g.][]{Gar-Her2006}  just at the end of the AGB evolution. The high
temperature annealing may lead to crystallization of amorphous silicates, while
carbon, again allowed to be dredged-up to the surface, may be used for the
PAH formation.

In our sample of GB PNe, we see an overwhelming presence of crystalline
silicates \citep[see also][]{Gutenkunst2008}. This might be explained by the
existence of long-lived O-rich disk (primordial or circumbinary) around CSs of
PNe as proposed in scenarios {\bf ii)} and {\bf iii)}.
However, these explanations appear to be inconsistent with the existing observations of AGB stars in the GB,
which in general do not show any evidence of crystalline silicates
\citep{Vanhollebeke2007}. In addition, \citet{Miszalski2008}
reported that out of 300 GB PNe only some dozen (about
10-15\%) display signatures of periodic variability (interpreted as due to binarity) in the
OGLE-III data. The lack of evidence for a dominant
binarity also excludes more exotic mechanisms, such as
these proposed in scenario {\bf iv}).
Scenario {\bf v}) is also not applicable to the GB PNe,
since the current AGB population in the Galactic Bulge
consists of mostly low-mass stars \citep[see e.g.][and references therein]{Uttenthaler_etal_2007}.

To elucidate the validity of  the scenario {\bf i}), it is important to
take into account the higher (on average) metallicity of the GB population
compared with the rest of Galaxy \citep{Chiappini2008}. Low-mass O-rich AGB
stars in the GB, with limited efficiency in the third dredge-up, do not succeed
in bringing enough carbon into the envelope to produce C-rich AGB stars in a
high-metallicity environment. Hence, it seems very likely that the formation of C-rich
PNe (so numerous in the GB) must originate in a final He-shell instabillity (probably
violent), which exposes the He- and C-rich layers only when the AGB phase is
terminated.

The large mass loss expected in GB PNe at the end of the
AGB due to the higher metallicity compared with the average Galactic metallicity
(and thus higher amount of dust) may facilitate the mixing/removal of the remaining hydrogen from
the stellar envelope leading to the formation of [WC] CSs. In addition,  this
violent process may be connected to the operation of strong shocks, which  can
increase the temperature in circumstellar shells leading to crystallization  of
silicates. Therefore, we suggest that process {\bf i}) proposed above appear
to be the most plausible scenario in explaining the
formation of C-rich PNe with mixed  chemistry in the GB.

The detection of dual-dust chemistry in the majority of the GB PNe
constitutes the first evidence that this phenomenon is not restricted to
late-type WCPNe, as previously suggested. We suggest that the
dual-dust chemistry in the GB PNe is related to the low
mass and high metallicity of their precursors. The missing population of C-rich AGB stars in
the GB and the overall lack of crystalline silicates around GB AGB stars,
suggest that the formation of PNe with dual-chemistry
may be due to a final He-shell instability that leads to
a simultaneous change in chemistry (PAH formation) and
crystallization of  pre-existing silicates. However, other possibilities for dual-dust
chemistry cannot be excluded in other than GB environment. The problem clearly deserves
further theoretical analysis and needs to take into account the new observational
constraints provided here.

\begin{acknowledgements}
We acknowledge support from the Faculty of the European Space Astronomy Centre
(ESAC) and from the Comunidad de Madrid PRICIT project S-0505/ESP-0237
(ASTROCAM). RSz acknowledges support from grants N203 393334 and N203 0661 33.
We thank to Slawek G\'orny for his useful comments and help during the
preparation of this manuscript. We are also grateful to the referee, Dr. M.
Busso, whose comments allowed us to improve the manuscript.
\end{acknowledgements}

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Arean för den av öarna koordinaterna pekar på är  hektar och dess största längd är 140 meter i öst-västlig riktning. Närmaste större samhälle är Vasa,  km öster om Sälgen.

Klimat
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Källor

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text:
\section{Introduction}

Let $p$ be a prime number and $t$ be a positive integer. Let $\Fq$ be a finite field with $q$ elements, where $q=p^n$.  For $\vec{a}=(a_1,\dots,a_s)\in\Fq^s$, $\vec{d}=(d_1,\dots,d_s)\in\Z_+^s$, $\vec{t}=(t_1,\dots,t_s)\in\Z_+^s$ and $b\in\Fq$, let $N_s(\vec{a},\vec{d},\vec{t},q,b)$ be the number of solutions of the diagonal equation
\begin{equation}\label{item300}
a_1 x_1^{d_1}+\cdots+a_s x_s^{d_s}=b
\end{equation}
where $x_i\in\Fpti$. In the case $t_1=\dots=t_s=n$,  Weil~\cite{weil1949numbers} and Hua and Vandiver~\cite{hua1949characters} independently showed that $N_s(\vec{a},\vec{d},q,b)$ can be expressed in terms of character sums. In particular, Weil's result implies that
\begin{equation}\label{item301}
|N_s(\vec{a},\vec{d},\vec{t},q,0)-q^{s-1}|\le I(d_1,\dots,d_s)(q-1)q^{(s-2)/2}
\end{equation}
where $I(d_1,\dots,d_s)$ is the number of $s$-tuples $(y_1,\dots,y_s)\in\Z^n$, with $1\leq y_i\leq d_i-1$ for all $i=1,\dots,s$, such that
\begin{equation}\label{item302}
\frac{y_1}{d_1}+\dots+\frac{y_s}{d_s}\equiv 0\pmod{1}.
\end{equation}
 The study of solutions of diagonal equations played an essential role in the statement of Weil's conjectures for algebraic varieties and in the development of the algebraic geometry. The number of solutions of diagonal equations have been extensively studied in the last few decades \cite{cao2007factorization,baoulina2010number,hou2009certain,sun1996number,qi1997diagonal,oliveira2019rational,baoulina2016class,zhou2019counting}. In many cases, the authors present a formula for the number of solutions of equations whose exponents satisfy certain natural restrictions. The case where $q$ is a square provides families of diagonal equations whose number of solutions can be obtained by means of simple parameters ~\cite{wolfmann1992number,cao2016number}. Recently, we presented \cite{oliveira2020diagonal} a formula for the number of solutions of Eq.~\eqref{item300} in the case where $t_1=\dots=t_s=n$, $q=p^{2n}$ and  there exists a divisor $r$ of $n$ such that $d_i|(p^r+1)$ for all $i=1,\dots,s$. Most notably, in \cite{oliveira2020diagonal} we studied the number of solutions of \eqref{item300} in order to find those diagonal equations whose number of solutions attains the bound \eqref{item301}.  For more results concerning diagonal equations over finite fields, see Section 7.3 in \cite{Panario} and the references therein.

 In \cite{reis2020counting}, the author studies a class of suitable generalized diagonal equations with restricted solutions sets. Inspired by the results of \cite{reis2020counting} and by the results obtained recently in \cite{oliveira2020diagonal}, this paper's aim is to generalize the results of~\cite{oliveira2020diagonal} to diagonal equations with restrict solution sets.
 In order to do that, we employ some well-known results on quadratic forms over finite dimensional vector spaces over a finite field. Also, we propose some open problems and present conjectures on the number of solutions of affine varieties.

 As follows, we state our main results and some important remarks. Throughout the paper, unless otherwise stated,  $q=p^{2n}$ for some positive integer $n$ and an odd prime $p$. Let $n$ be a positive integer. For $t$ a divisor of $n$, let $\tr_{p^n,p^{t}}$ denote the trace function from $\F_{p^n}$ to $\F_{p^t}$. Throughout the paper, unless otherwise stated, we let $\vec{a}\in\Fq^s$, $t=(2t_1,\dots,2t_s)$ with $t_i|n$ for all $i=1,\dots,s$. Since we are interested in the number of solutions of Eq.~\eqref{item300}, by a simple change of variables, we may assume without loss of generality that $d_i$ is a divisor of $p^{2t_i}-1$ for all $i=1,\dots,s$.
 \begin{definition}\label{item312}
 	Let $a\in\Fq$ and let $d,t,r$ be positive integers such that  and $r|t$, $t|n$ and $d|(p^{2t}-1)$. We set $m=\tfrac{p^{2t}-1}{d}$, $u=\tfrac{p^r+1}{d}$ and $\varepsilon=(-1)^{t/{r}}$. Let $T(x)=\tr_{q,p^{2t}}(x)$.
 	\begin{enumerate}[label=(\alph*)]
 		\item For $c\in\Fq$, let
 		$${\small \Delta(a,d,t,r,c)=\begin{cases}
 			p^{t}, &\text{ if }T(ca)=0;\\
 			\varepsilon(1-d), &\text{ if }T(ca)^{m}=\varepsilon^{u};\\
 			\varepsilon, &\text{ if }T(ca)^{m}\not\in\{0, \varepsilon^{u}\};\\
 			\end{cases}}$$
 		\item Let $\Lambda(a,t)=\{c\in\Fq:T(ca)=0\}$.
 	\end{enumerate}
 \end{definition}

 The main results of the paper are the following.

 \begin{theorem}\label{item303}
 	Assume that for each $i$, with $i=1,\dots,s$, there exists a divisor $r_i$ of $t_i$ such that $d_i|(p^{r_i}+1)$. Then
 	$$N_s(\vec{a},\vec{d},\vec{t},q,0)=p^{t_1+\cdots+t_s-2n}\sum_{c\in\Fq^*}\prod_{i=1}^{s} \Delta(a_i,d_i,t_i,r_i,c),$$
 	where $\Delta(a_i,d_i,t_i,r_i,c)$ is defined as in Definition~\ref{item312}.

 \end{theorem}

\begin{remark}
	As a direct consequence of Theorem~\ref{item303}, we have that $p^{t_1+\cdots+t_s-2n}$ divides $N_s(\vec{a},\vec{d},\vec{t},q,0)$. Therefore, if $N_s(\vec{a},\vec{d},\vec{t},q,0)>0$, then $N_s(\vec{a},\vec{d},\vec{t},q,0)\geq p^{t_1+\cdots+t_s-2n}$.
\end{remark}

 \begin{corollary}\label{item309}
	Let $b\in\Fq^*$ and suppose $d_1=\dots=d_s=d$. Assume that for each $i$, with $i=1,\dots,s$, there exists a divisor $r_i$ of $t_i$ such that $d|(p^{r_i}+1)$. Let $t_l$ be the least integer such that $p^{t_l}\equiv -1\pmod{d}$. Then
	$$N_s(\vec{a},\vec{d},\vec{t},q,b)=\frac{p^{t_1+\cdots+t_s-2n}}{p^{2t_l}-1}\sum_{c\in\Fq^*}\left(\Delta(-b,d,t_l,t_l,c)p^{t_l} -1\right)\prod_{i=1}^{s}\Delta(a_i,d,t_i,r_i,c),$$
	where $\Delta(-b,d,t_l,t_l,c)$ and $\Delta(a_i,d,t_i,r_i,c)$ are defined as in Definition~\ref{item312}.
\end{corollary}

Many results in the literature (for example Theorem 1 of \cite{wolfmann1992number}, Theorem 2.9 of \cite{cao2016number} and Theorem 2.3 of \cite{oliveira2020diagonal}) can be obtained by a straight employment of Theorem~\ref{item303} and Corollary~\ref{item309}, which means that the results of this paper are far more general than the known results. The case where $d_1=\dots=d_s=2$ and $n$ is even is covered by Corollary~\ref{item309}. More about the case where $d_1=\dots=d_s=2$ is discussed in Section~\ref{item334}.

\begin{corollary}\label{item313}  Assume that for each $i$, with $i=1,\dots,s$, there exists a divisor $r_i$ of $t_i$ such that $d|(p^{r_i}+1)$. Then
	$$\left|\frac{N_s(\vec{a},\vec{d},\vec{t},q,0)}{ p^{t_1+\cdots+t_s-2n}}-|\Lambda|p^{t_1+\cdots+t_s}\right|\leq \sum_{c\in\Fq\backslash\Lambda}\ \prod_{i\in \nu(c)} p^{t_i}\prod_{i\not\in \nu(c)}(d_i-1),$$
	where $\Lambda=\cap_{i=1}^{s}\Lambda(a_i,t_i)$ and $\nu(c)=\{i\in\{1,\dots,s\}:c\in\Lambda(a_i,t_i)\}$. In particular, if $\Lambda(a_i,t_i)\cap \Lambda(a_j,t_j)=\{0\}$ for all $1\leq i< j \leq s$ and
	$$\prod_{i=1}^s(d_i-1)\Bigg(\sum_{i=1}^s\bigg(\frac{1}{p^{t_i}(d_i-1)}-\frac{1}{p^{2t_i}}\bigg)+1+\frac{s-1}{q}\Bigg)<p^{t_1+\cdots+t_s-2n},$$
	 then there exists one solution for Equation~\ref{item300}. In this case, there exist at least $p^{t_1+\cdots+t_s-2n}$ solutions.
\end{corollary}

One can obtain bounds for the case $b\neq 0$ and $d_1=\dots=d_s$ from Corollary~\ref{item309}. In particular, tight bounds can be obtained in case where $s=2$, as we will see in Section~\ref{item325}.

\begin{corollary}\label{item311}
	 Let $t_l$ be the least integer such that $p^{t_l}\equiv -1\pmod{d}$. The number of solutions of the equation $a_1 x_1+\dots+a_s x_s=b$ with $x_i\in\F_{p^{2t_i}}$ is given by
	$$\frac{\lambda_b p^{2t_1+\cdots+2t_s+2t_l-2n}-\lambda p^{2t_1+\cdots+2t_s-2n}}{p^{2t_l}-1},$$
	where $\lambda:=|\cap_{i=1}^{s}\Lambda(a_i,t_i)|$ and $\lambda_b:=|\cap_{i=1}^{s}\Lambda(a_i,t_i)\cap\Lambda(-b,t_l)|$.
\end{corollary}

\begin{remark}
	In the particular case $b=0$, it follows that $\lambda_0=\lambda$ in Corollary~\ref{item311} and then the number of solutions of the equation $a_i x_1+\dots+a_s x_s=0$ with $x_i\in\F_{p^{2t_i}}$ equals $\lambda p^{2t_1+\cdots+2t_s-2n}$.
\end{remark}

The paper is organized as follows. Section~\ref{item332} provides preliminary results which will be used to prove the main results. In Section~\ref{item333}, we prove the main results of the paper. In Section~\ref{item325}, we study diagonal equations with two variable and give a bound for its number os solutions, generalizing Hasse-Weil bound for this class of equations. Section~\ref{item334} discuss the case $d_1=\dots=d_s=2$. Finally, in Section~\ref{item335} we provide some final considerations and open problems.

\section{On the number of solutions of $\tr_{p^{2n},p}(ax^d)=\lambda$}\label{item332}

In this section we provide the tools from quadratic forms over finite fields which will be used throughout the paper. We use the ideas from Wolfman~\cite{wolfmann1989number} in order to generalize his results. The following result is well known (see \cite{Lidl}).

\begin{lemma}\label{item304}
	Let $\Phi$ be a quadratic form on a $k$ dimensional vector space $E$ over $\Fq$, $\Psi$ the associated symmetric bilinear form and $v$ the dimension of the kernel of $\Psi$. For $\lambda\in\F_q$, let $S_\lambda$ the number of solutions in $E$ of $\Phi(x)=\lambda$. If $k=2t$ and $v=2s$, then there exists $D\in\{-1,0,1\}$ such that
	$$S_\lambda=\begin{cases}
	q^{2t-1}+Dq^{t+s-1}(q-1),&\text{ if }\lambda=0;\\
	q^{2t-1}-Dq^{t+s-1},&\text{ if }\lambda\neq0.\\
	\end{cases}$$
\end{lemma}

\begin{proposition}\label{item308}
	Let $q=p^{2n}$ and let $t$ and $r$ be integers such that $t|n$ and $r|t$. Let $a\in\Fq$. Then the map $\Phi$ from $\F_{p^{2t}}$ into $\Fp$ defined by $\Phi_a(x)=\tr_{q,p}(ax^{p^r+1})$ is a quadratic form. Let $\hat{a}=\tr_{q,p^{2t}}(a)$ and $m=\tfrac{p^{2t}-1}{p^r+1}$. If $s$ and $D$ are integers as defined in Lemma~\ref{item304}, then
	\begin{enumerate}
		\item If $\hat{a}=0$, then $s=t$ and $D=1$;
		\item If $\hat{a}^m=(-1)^{t/r}$, then $s=r$ and $D=(-1)^{t/r-1}$;
		\item If $\hat{a}^m\not\in\{0,(-1)^{t/r}\}$, then $s=0$ and $D=(-1)^{t/r}$;
	\end{enumerate}
\end{proposition}

\begin{proof}
	We observe that $\Phi_a(x+y)=\Phi_a(x)+\Phi_a(y)+\tr_{q,p}(ax^{p^r}y+ay^{p^r}x)$ and $\tr_{q,p}(ax^{p^r}y+ay^{p^r}x)$ is a symmetric bilinear form.  Furthermore, if $\lambda\in\Fp$, then $\Phi_a(x)=\tr_{q,p}(ax^{p^r+1})$ and therefore $\Phi_a$ is a quadratic form. Now we compute the dimension of the kernel of $\tr_{q,p}(ax^{p^r}y+ay^{p^r}x)$. Let $\hat{a}=\tr_{q,p^{2t}}(a)$ and observe that
	$$\begin{aligned}
	\tr_{q,p}(ax^{p^r}y+ay^{p^r}x)&=\tr_{p^{2t},p}\left(\tr_{q,p^{2t}}(ax^{p^r}y+ay^{p^r}x)\right)\\
	&=\tr_{p^{2t},p}\left((x^{p^r}y+y^{p^r}x)\tr_{q,p^{2t}}(a)\right)\\
	&=\tr_{p^{2t},p}\left(x^{p^r}y\hat{a}\right)+\tr_{p^{2t},p}\left(y^{p^r}x \hat{a}\right)\\
	&=\tr_{p^{2t},p}\left(x^{p^r}y\hat{a}\right)+\tr_{p^{2t},p}\left(yx^{p^{2t-r}} \hat{a}^{p^{2t-r}}\right)\\
	&=\tr_{p^{2t},p}\left(y\left( x^{p^r}\hat{a}+x^{p^{2t-r}} \hat{a}^{p^{2t-r}}\right) \right),\\
	\end{aligned}$$
	which vanish for all $y$ if and only if $x^{p^r}\hat{a}+x^{p^{2t-r}} \hat{a}^{p^{2t-r}}=0$, which is equivalent to
	\begin{equation}\label{item305}
	x^{p^{2r}}\hat{a}^{p^r}+x \hat{a}=0.
	\end{equation}
	If $\hat{a}=0$, then $x^{p^r}\hat{a}+x^{p^{2t-r}} \hat{a}^{p^{2t-r}}=0$ for all $x\in\F_{p^{2t}}$ and so $s=t$. Otherwise, the nonzero solutions of Eq.~\ref{item305} are the elements $x\in\F_{q^t}$ such that $x^{p^{2r}-1}=-1\hat{a}^{1-p^r}$. The equation $x^{p^{2r}-1}=-1\hat{a}^{1-p^r}$ has solutions if and only if
	\begin{equation}\label{item306}
	\left(-1\hat{a}^{1-p^r}\right)^{(p^{2t}-1)/(p^{2r}-1)}=1
	\end{equation}
	 and, in the affirmative case, there exist $p^{2r}-1$ distinct solutions. We observe that Eq.~\ref{item306} is equivalent to $\hat{a}^{\frac{p^{2t}-1}{p^r+1}}=(-1)^{t/r}$ and this proves our statements for kernel's dimension. Now, we compute the value of $d$. Let $L$ be the multiplicative subgroup of order $p^{r}+1$ of $\Fq^*$. If $c\in\F_{p^{2t}}$ is an element such that $\Phi_a(c)=\lambda$ for some $\lambda\in \F_p^*$, then $\Phi_a(cl)=\lambda$ for all $l\in L$. Therefore, Lemma~\ref{item304} entails that
	 $$p^{2t-1}\equiv Dp^{t+s-1}\pmod{p^r+1},$$
	 which implies that $D=(-1)^{(t-s)/r}$, since we have already proved that $r|(t-s)$. This completes the proof of our proposition.
\end{proof}

\begin{remark}
	We can change $p$ by a power of prime without any further arguments in the proof. We use $p$ in our statements to make the proofs of our main results simpler.
\end{remark}

\begin{theorem}\label{item307}
	Let $q=p^{2n}$ and let $t, r$ and $d$ be integers such that $t|n$,  $r|t$ and $d|(p^r+1)$. Let $a\in\F_{q}^*$, $\lambda\in\Fp$ and let $N_\lambda$ denote the number of solutions of $\tr_{q,p}\big(ax^{d}\big)=\lambda$ over $\F_{p^{2t}}$. Set $\hat{a}=\tr_{q,p^{2t}}(a)$, $k=\tfrac{p^{2t}-1}{d}$, $\varepsilon=(-1)^{t/r}$ and $u=\tfrac{p^r+1}{d}$. Then
	\begin{enumerate}[label=(\alph*)]
		\item $\hat{a}=0$ and $\lambda= 0$, $N_\lambda=p^{2t}$;
		\item $\hat{a}=0$ and $\lambda\neq 0$, $N_\lambda=0$;
		\item $\hat{a}^k\not\in \{0,\varepsilon^u\}$ and $\lambda= 0$, $N_\lambda=p^{2t-1}+\varepsilon p^{t-1}(p-1)$;
		\item $\hat{a}^k\not\in \{0,\varepsilon^u\}$ and $\lambda\neq 0$, $N_\lambda=p^{2t-1}+\varepsilon p^{t-1}$;
		\item $\hat{a}^k= \varepsilon^u$ and $\lambda\neq 0$, $N_\lambda=p^{2t-1}-\varepsilon p^{t-1}(d-1)(p-1)$;
		\item $\hat{a}^k= \varepsilon^u$ and $\lambda= 0$, $N_\lambda=p^{2t-1}-\varepsilon p^{t-1}(d-1)$.
	\end{enumerate}
\end{theorem}

\begin{proof} Let $\alpha$ be a primitive element of $\F_{p^{2t}}^*$. We observe that
	$$\big\{x\in\F_{p^{2t}}:\tr_{q,p}\big(ax^d\big)=\lambda\big\}=\bigcup_{\ell=0}^{\frac{p^r+1}{d}-1}\big\{x\in\F_{p^{2t}}:\tr_{q,p}\big(a\alpha^{\ell d} x^{p^r+1}\big)=\lambda\big\},$$
	where which element of the left-hand set has $\tfrac{p^r+1}{d}$ representatives in the right-hand side. In particular,
	\begin{equation}\label{item209}
	N_\lambda=|\{x\in\F_{p^{2t}}:\tr_{q,p}\big(ax^d\big)=\lambda\}|=\frac{d}{p^r+1}\left|\bigcup_{\ell=0}^{\frac{p^r+1}{d}-1}\{x\in\F_{p^{2t}}:\tr_{q,p}\big(a\alpha^{\ell d} x^{p^r+1}\big)=\lambda\}\right|.
	\end{equation}
	For $\ell=0,\dots,\tfrac{p^r+1}{d}-1$, we set $\hat{a}_\ell=\tr_{q,p^{2t}}(a\alpha^{\ell d})$ and observe that $\tr_{q,p^{2t}}(a\alpha^{\ell d})=\alpha^{\ell d}\tr_{q,p^{2t}}(a)=\alpha^{\ell d} \hat{a}$, so that $\hat{a}_\ell=0$ if and only if $\tr_{q,p^{2t}}(a)=0$. In this case, Proposition~\ref{item308} and Eq.~\eqref{item209} implies that $N_0=q^{2t}$ and $N_\lambda=0$ if $\lambda\neq 0$. Let $m=\tfrac{p^{2t}-1}{p^r+1}$ and $\varepsilon=(-1)^{t/r}$. We split the remaining part of the proof into two cases:
	\begin{itemize}
		\item If there exists no $j\in\{0,\dots,\tfrac{p^r+1}{d}-1\}$ such that $\hat{a}_j^m=(-1)^{t/r}$, then Proposition~\ref{item308} and Eq.~\eqref{item209} entail that
		$$N_\lambda=\begin{cases}
		p^{2t-1}+\varepsilon p^{t-1}(p-1),&\text{ if }\lambda=0;\\
		p^{2t-1}-\varepsilon p^{t-1},&\text{ if }\lambda\neq 0.\\
		\end{cases}
		$$
		\item If there exists some $j\in\{0,\dots,\tfrac{p^r+1}{d}-1\}$ such that $\hat{a}_j^m=(-1)^{t/r}$, then it is easy to verify that $\hat{a}_i^m\neq (-1)^{t/r}$ for all $i\in\{0,\dots,\tfrac{p^r+1}{d}-1\}\backslash\{j\}$, so that a straightforward computation using Proposition~\ref{item308} and Eq.~\eqref{item209} shows that
		$$N_\lambda=\begin{cases}
		p^{2t-1}-\varepsilon p^{t-1}(d-1)(p-1),&\text{ if }\lambda=0;\\
		p^{2t-1}-\varepsilon p^{t-1}(d-1),&\text{ if }\lambda\neq 0.\\
		\end{cases}
		$$
	\end{itemize}
	 Our assertion follows by observing that there exists $j\in\{0,\dots,\tfrac{p^r+1}{d}-1\}$ such that $\hat{a}_j^m=(-1)^{t/r}$ if and only if $\hat{a}^{(p^{2t}-1)/d}=\varepsilon^{(p^r+1)/d}$.
\end{proof}

\section{Proof of the main results}\label{item333}
In this section, we prove our counting results. To prove Theorem \ref{item303}, we will follow the main ideas of Wolfmann~\cite{wolfmann1992number}. To this end, let $a\in\Fq$ and let $\psi_a(x)=\exp\left((2\pi i)\tr_{q,p}(ax)/p\right)$ be an additive character. We have the following known results.

\begin{lemma}\label{item324}
	Let $\vec{a}=(a_1,\dots,a_s)\in\Fq^s$ and $\vec{d}=(d_1,\dots,d_s)\in\Z_+^s$ such that $d_i|(q-1)$ for all $i=1,\dots,s$. Then
	$$N_s(\vec{a},\vec{d},\vec{t},q,b)=q^{-1}\sum_{c\in\scalebox{0.7}{$\displaystyle \Fq$}} \psi_c(-b)\prod_{i=1}^{s}S_{i}(c),$$
	where $S_{i}(c):=\sum_{x\in\scalebox{0.7}{$\displaystyle \F_{p^{2t_i}}$}}\psi_{ca_i}\left(x^{d_i}\right)$.
\end{lemma}
\begin{proof}
	It follows by a argument similar to the proof of Proposition 1 of \cite{wolfmann1992number}.
\end{proof}

\begin{lemma}\label{item323}
	Let $q=p^{2n}$, $c\in\Fq$, and $\vec{a},\vec{d}$ and $S_{i}(c)$ as defined in Lemma~\ref{item324}. Suppose that there exists a divisor $r_i$ of $t_i$ such that $d_i|(p^{r_i}+1)$ for each $i=1,\dots,s$. Let $\varepsilon_i=(-1)^{t_i/r_i}$, $\epsilon_i=\varepsilon_i^{(p^{r_i}+1)/d_i}$ and $\hat{a}_i=\tr_{q,p^{2t_i}}(ca_i)$. Then, for each $i=1,\dots,s$, we have that
	\begin{enumerate}[label=(\alph*)]
		\item If $\hat{a}_i=0$, then $S_i(c)=p^{2t_i}$;\\
		\item If $\hat{a}_i^{(p^{2t_i}-1)/d_i}=\epsilon_i$, then $S_i(c)=-\varepsilon_i(d_i-1)p^{t_i}$;\\
		\item If $\hat{a}_i^{(p^{2t_i}-1)/d_i}\not\in\{0, \epsilon_i\}$, then $S_i(c)= \varepsilon_i\, p^{t_i} $.\\
	\end{enumerate}
\end{lemma}

\begin{proof}
	We observe that
	\begin{equation}\label{item328}
	S_{i}(c)=\sum_{\lambda\in\Fp} \exp\left(\frac{2\pi i\lambda}{p}\right)N_\lambda,
	\end{equation}
	where $N_\lambda$ is defined as in Lemma~\ref{item307}. Our result follows directly from Eq.~\eqref{item328} and Theorem~\ref{item307}.
\end{proof}

From Lemmas \ref{item324} and \ref{item323}, we are able to prove Theorem \ref{item303}.

\subsection{Proof of Theorem \ref{item303}} Since $b=0$, it follows from Lemma~\ref{item324} that
	$$N_s(\vec{a},\vec{d},\vec{t},q,0)=q^{-1}\sum_{c\in\scalebox{0.7}{$\displaystyle \Fqdo$}} \prod_{i=1}^{s}S_{i}(c).$$
	Let $\varepsilon_i=(-1)^{t_i/r_i}$, $m_i=\tfrac{p^{2t_i}-1}{d_i}$ and $u_i=\tfrac{p^{r_i}+1}{d_i}$. Let $T_i(x)=\tr_{q,p^{2t_i}}(x)$. By Lemma~\ref{item323}, it follows that $S_{i}(c)=p^{t_i}\Delta(a_i,d_i,t_i,r_i,c)$, where
	$${\small \Delta(a_i,d_i,t_i,r_i,c)=\begin{cases}
		p^{t_i}, &\text{ if }T_i(ca_i)=0;\\
		\varepsilon_i(1-d_i), &\text{ if }T_i(ca_i)^{m_i}=\varepsilon_i^{u_i};\\
		\varepsilon_i, &\text{ if }T_i(ca_i)^{m_i}\not\in\{0, \varepsilon_i^{u_i}\}.\\
		\end{cases}}$$
	This completes the proof of our assertion.	$\hfill\qed$

\begin{definition}
	For $f\in\Fq[x_1,\dots,x_s]$ and $\vec{t}=(t_1,\dots,t_s)$ with $t_i|n$ for all $i=1,\dots,s$, let
	$$V_{\vec{t}\,}(f(x_1,\dots,x_s)=0)=\{(x_1,\dots,x_s)\in\F_{t_1}\times\dots\times\F_{t_s}:f(x_1,\dots,x_s)=0\},$$
	the set of zeros of $f$. For $i\in\{1,\dots,s\}$, let
	$$V_{\vec{t},\, x_i}(f(x_1,\dots,x_s)=0)=\{(x_1,\dots,x_s)\in\F_{t_1}\times\dots\times\F_{t_s}:f(x_1,\dots,x_s)=0\text{ and }x_i\neq 0\}.$$
\end{definition}

\subsection{Proof of Corollary~\ref{item309}}
Let $b\in\Fq^*$ and let $x_1,\dots,x_s,y$ denote indeterminates. Let $\vec{t_0}=(2t_1,\dots,2t_s,2t_l)$ and $\vec{t}=(2t_1,\dots,2t_s)$. Let
 $$\psi: V_{\vec{t_0},\, y}(a_1 x_1^{d}+\cdots+a_s x_s^{d}=by^{d})\rightarrow V_{\vec{t}\, }(a_1 x_1^{d}+\cdots+a_s x_s^{d}=b)$$
 be the function defined by
$$(x_1,\dots,x_s,y)\mapsto(x_1/y,\dots,x_s/y).$$
Since $t_l|t_i$ for all $i=1,\dots,s$, it follows that $\psi$ is well-defined. It is direct to verify that $\psi$ is a $(p^{2t_l}-1)$-to-one function and then
\begin{equation}\label{item314}
|V_{\vec{t_0},\, y}(a_1 x_1^{d}+\cdots+a_s x_s^{d}=by^{d})|=(p^{2t_l}-1)|V_{\vec{t}\, }(a_1 x_1^{d}+\cdots+a_s x_s^{d}=b)|.
\end{equation}
We observe that $V_{\vec{t_0},\, y}(a_1 x_1^{d}+\cdots+a_s x_s^{d}=by^{d})$ is iqual to
$$ V_{\vec{t_0}\,}(a_1 x_1^{d}+\cdots+a_s x_s^{d}=by^{d})\backslash \left(V_{\vec{t_0}\,}(a_1 x_1^{d}+\cdots+a_s x_s^{d}=by^{d})\cap\{(x_1,\dots,x_s,y):y=0\}\right)$$
and
$$|V_{\vec{t_0}\,}(a_1 x_1^{d}+\cdots+a_s x_s^{d}=by^{d})\cap\{(x_1,\dots,x_s,y):y=0\}|=|V_{\vec{t}\, }(a_1 x_1^{d}+\cdots+a_s x_s^{d}=0)|,$$
so that
{\small\begin{equation}\label{item315}
|V_{\vec{t_0},\, y}(a_1 x_1^{d}+\cdots+a_s x_s^{d}=by^{d})|=|V_{\vec{t_0}\,}(a_1 x_1^{d}+\cdots+a_s x_s^{d}=by^{d})|-|V_{\vec{t}\, }(a_1 x_1^{d}+\cdots+a_s x_s^{d}=0)|.
\end{equation}}
Let $\vec{a_0}=(a_1,\dots,a_s,b)$ and $\vec{d_0}=(d,\dots,d)\in\Z^{s+1}$. Since $N_s(\vec{a},\vec{d},\vec{t},q,b)=|V_{\vec{t}\, }(a_1 x_1^{d}+\cdots+a_s x_s^{d}=b)|$, $N_s(\vec{a},\vec{d},\vec{t},q,0)=V_{\vec{t}\, }(a_1 x_1^{d}+\cdots+a_s x_s^{d}=0)$ and $N_{s+1}(\vec{a_0},\vec{d_0},\vec{t_0},q,0)=V_{\vec{t_0}\,}(a_1 x_1^{d}+\cdots+a_s x_s^{d}=by^{d})$, our result follows from Eq.~\eqref{item314} and \eqref{item315} and Theorem~\ref{item303}.	$\hfill\qed$\\

\begin{remark}\label{item322}
	We observe that the condition on $t_l$ in Corollary~\ref{item309} can be relaxed to $p^{t_l}\equiv -1\pmod{d}$ and $t_l|t_i$ for all $i$ and the same proof works.
\end{remark}

\subsection{Proof of Corollary~\ref{item313}}
 Let $\Lambda=\cap_{i=1}^s\Lambda(a_i,t_i)$. By Theorem~\ref{item303}, we have that
 $$\begin{aligned}N_s(\vec{a},\vec{d},\vec{t},q,0)p^{2n-t_1-\cdots-t_s}&=\sum_{c\in\Fq^*}\prod_{i=1}^{s} \Delta(a_i,d_i,t_i,r_i,c)\\
 &=\sum_{c\in\Lambda}\prod_{i=1}^{s} \Delta(a_i,d_i,t_i,r_i,c)+\sum_{c\in\Fq\backslash\Lambda}\prod_{i=1}^{s} \Delta(a_i,d_i,t_i,r_i,c)\\
 &=|\Lambda|p^{t_1+\dots+t_s}+\sum_{c\in\Fq\backslash\Lambda}\prod_{i=1}^{s} \Delta(a_i,d_i,t_i,r_i,c),\\
 \end{aligned}$$
 and then
 \begin{equation}\label{item316}
 \begin{aligned}\left|N_s(\vec{a},\vec{d},\vec{t},q,0)p^{2n-t_1-\cdots-t_s}-|\Lambda|p^{t_1+\dots+t_s}\right|\leq \sum_{c\in\Fq\backslash\Lambda}\prod_{i=1}^{s} \Delta(a_i,d_i,t_i,r_i,c).
 \end{aligned}
 \end{equation}
 By the definition of $\Delta(a_i,d_i,t_i,r_i,c)$, it follows that
 $$|\Delta(a_i,d_i,t_i,r_i,c)|\leq \begin{cases}
 p^{t_i},&\text{ if }c\in\Lambda(a_i,t_i);\\
 d_i-1,&\text{ if }c\not\in\Lambda(a_i,t_i)\\
 \end{cases}$$
 and therefore the Triangle Inequality on Eq.~\eqref{item316} implies that
 \begin{equation}\label{item317}
 \left|\frac{N_s(\vec{a},\vec{d},\vec{t},q,0)}{ p^{t_1+\cdots+t_s-2n}}-|\Lambda|p^{t_1+\cdots+t_s}\right|\leq \sum_{c\in\Fq\backslash\Lambda}\ \prod_{i\in \nu(c)} p^{t_i}\prod_{i\not\in \nu(c)}(d_i-1),
 \end{equation}
 where $\nu(c):=\{i\in\{1,\dots,s\}:c\in\Lambda(a_i,t_i)\}$. This completes the first part of our result. Assume that $\Lambda(a_i,t_i)\cap \Lambda(a_j,t_j)=\{0\}$ for all $1\leq i< j \leq s$. In this case, we have that
 {\small$$\sum_{c\in\Fq\backslash\Lambda}\ \prod_{i\in \nu(c)} p^{t_i}\prod_{i\not\in \nu(c)}(d_i-1)
 	\leq \sum_{j=1}^s |\Lambda(a_j,t_j)|\cdot p^{t_j}\prod_{i\neq j}(d_i-1)+\Bigg(q-1-\sum_{j=1}^s(|\Lambda(a_j,t_j)|-1)\Bigg)\prod_{i=1}^s(d_i-1).$$}
 Since $|\Lambda(a_j,t_j)|=q/p^{2t_i}$, it follows that
 {\small\begin{equation}\label{item318}
 	\sum_{c\in\Fq\backslash\Lambda}\ \prod_{i\in \nu(c)} p^{t_i}\prod_{i\not\in \nu(c)}(d_i-1)\leq q\prod_{i=1}^s(d_i-1)\Bigg(\sum_{i=1}^s\bigg(\frac{1}{p^{t_i}(d_i-1)}-\frac{1}{p^{2t_i}}\bigg)+1+\frac{s-1}{q}\Bigg)
 	\end{equation}}
 Therefore, Eq.~\eqref{item317} and \eqref{item318} imply that
 $$\frac{N_s(\vec{a},\vec{d},\vec{t},q,0)}{ p^{t_1+\cdots+t_s-2n}}\geq p^{t_1+\cdots+t_s}-  q\prod_{i=1}^s(d_i-1)\Bigg(\sum_{i=1}^s\bigg(\frac{1}{p^{t_i}(d_i-1)}-\frac{1}{p^{2t_i}}\bigg)+1+\frac{s-1}{q}\Bigg),$$
 which is bigger than zero whenever
 $$\prod_{i=1}^s(d_i-1)\Bigg(\sum_{i=1}^s\bigg(\frac{1}{p^{t_i}(d_i-1)}-\frac{1}{p^{2t_i}}\bigg)+1+\frac{s-1}{q}\Bigg)<p^{t_1+\cdots+t_s-2n}.$$
 This completes the proof of the Corollary. $\hfill\qed$\\

 \subsection{Proof of Corollary~\ref{item311}}

 Let $d_1=\dots=d_s=1$. Applying Corollary~\ref{item309} by taking $r_i=t_i$, it follows that
 \begin{equation}\label{item319}
 N_s(\vec{a},\vec{d},\vec{t},q,b)=\frac{p^{t_1+\cdots+t_s-2n}}{p^{2t_l}-1}\sum_{c\in\Fq^*}\left(\Delta(-b,1,1,1,c)p^{t_l} -1\right)\prod_{i=1}^{s}\Delta(a_i,1,t_i,t_i,c),
 \end{equation}
 where
 $${\small \Delta(a_i,1,t_i,t_i,c)=\begin{cases}
 	p^{t_i}, &\text{ if }\tr_{q,p^{2t_i}}(ca_i)=0;\\
 	0, &\text{ if }\tr_{q,p^{2t_i}}(ca_i)\neq 0\\
 	\end{cases}}$$
for all $i=1,\dots,s$ and
  $${\small \Delta(-b,1,t_i,t_i,c)=\begin{cases}
 	p^{t_i}, &\text{ if }\tr_{q,p^{2t_l}}(cb)=0;\\
 	0, &\text{ if }\tr_{q,p^{2t_l}}(cb)\neq 0.\\
 	\end{cases}}$$
 We set $\lambda=|\cap_{i=1}^{s}\Lambda(a_i,t_i)|$ and $\lambda_b=|\cap_{i=1}^{s}\Lambda(a_i,t_i)\cap\Lambda(b,t_l)|$ and then our result follows from the definition of $\Lambda(a,t)$ and Eq.~\eqref{item319}. $\hfill\qed$

\begin{comment}
\section{A Bound on the Number of Solutions of Diagonal Equations}
In this section, we consider a more general setting where the monomials $a_ix_i^{d_i}$ are replaced by polynomials $f_i(x_i)\in\Fq[x]$. In order to do so, we recall a well-known application of Weil's bound on character sums.

\begin{lemma}\cite[Theorem 5.38]{Lidl}\label{item310}
	Let $f\in\Fq[x]$ be of degree $d\geq 1$ with $\gcd(d,q)=1$ and let $\Psi$ be an additive character of $\Fq$. Then
	$$\sum_{c\in\Fq}\Psi(c)\leq (d-1)\sqrt{q}.$$
\end{lemma}

From this result, we are able to estimate the number of solutions of the diagonal equations with restricted solution sets.

\begin{theorem}
	Let $q=p^n$ and let $t_1,\dots,d_s$ be integers such that $t_i|n$. For each $i=1,\dots,s$, let $f_i\in\Fq[x]$ be a polynomial of degree $d_i$. Let $N$ denote the number of solutions of the diagonal equation $f_1(x_i)+\dots+f_s(x_s)=b$ with $x_i\in\F_{p^{t_i}}$ for every $i$. Then
	$$N\leq \prod_{i=1}^s(d_i-1)p^{t_i/2}$$
\end{theorem}

\end{comment}

\section{The case $s=2$}\label{item325}

  For a curve $\Ca$ over $\Fq$, we denote by $M_n(\Ca)$ the number of rational points of $\Ca$ over $\Fqn$. For an irreducible non-singular curve $\Ca$ over $\Fq$, the well-known Riemann Hypothesis \cite[Theorem $3.3$]{moreno1993algebraic} states that the number of rational points on a curve $\Ca$ over $\Fqn$ satisfies
$$M_n(\mathcal{C})=q^n+1-\sum\limits_{i=1}^{2g} \omega_i^n,$$
where $g$ denotes the genus of $\Ca$ and $|\omega_i|=\sqrt{q}$ for all $i$.
Also by this result, we have the well-known Hasse-Weil bound for the number of rational points on an irreducible non-singular curve over $\Fq$ of genus $g$, given by
\begin{equation}\label{item320}
|M_n(\mathcal{C})-q^n-1|\leq 2g\sqrt{q^n}.
\end{equation}
The curve $\Ca$ is called maximal over $\F_{q^2}$ if its number of points attains the Hasse-Weil upper bound, that is,
$$M_1(\Ca)=q^2+1+2gq,$$
where $g$ is the genus of $\Ca$. Similarly, a curve is called minimal over $\F_{q^2}$ if it attains the Hasse-Weil lower bound. There exist many families of maximal curves in the literature. In particular, a curve $\mathcal{J}$ with affine equation $x_1^{d}+x_2^{d}=1$ yields examples of maximal and minimal curves (e.g. see \cite{tafazolian2010characterization}). We observe that $M_n(\mathcal{J})=N_2(\vec{a},\vec{d},\vec{t},q,b)+c_0$ if $t_1=t_2=n$, where $c_0$ is the number of points at infinity on $\mathcal{J}$. Our aim in this section is to present a bound such as \eqref{item320} in the case the points on $\mathcal{C}$ has restricted solutions sets, replacing $M_n(\mathcal{C})$ by $N_2(\vec{a},\vec{d},\vec{t},q,b)$.

 \begin{corollary}\label{item321}
 	Let $q=p^{2n}$, $\vec{a}=(1,1)$, $\vec{d}=(d,d)$ and $\vec{t}=(t_1,t_2)$. Assume that there exists a divisor $r_1$ of $t_1$ and a divisor $r_2$ of $t_2$ such that $d|(p^{r_1}+1)$ and $d|(p^{r_2}+1)$. Let $t_l$ be the least integer such that $p^{t_l}\equiv -1\pmod{d}$. Then the number $N$ of points on the affine curve $x^d+y^d+1=0$ with $x\in\F_{q^{2t_1}}$ and $y\in\F_{q^{2t_2}}$ satisfies
	$$\big|N-|\Lambda|p^{2t_1+2t_2-2t}\big|\le (d-1)^2 p^{t_1+t_2}+(d-1)(p^{t_1}+p^{t_2}),$$
	where $\Lambda:=\Lambda(1,t_1)\cap\Lambda(1,t_2)$.
\end{corollary}
\begin{proof}
	Since $\Lambda(1,t_i)\subset\Lambda(1,t_l)$ for $i=1,2$, the result follows directly by Corollary~\ref{item309}.
\end{proof}

From here, pose the following problem.

\begin{problem}
	Can we obtain a bound similar to Hasse-Weil's bound for equations in two variables with restricted solutions sets?
\end{problem}

\section{The case $d_1=\dots=d_s=2$}\label{item334}

In the case where $d_1=\dots=d_s=2$, we can compute the number of solutions of diagonals equations over arbitrary finite fields, as we will see in this section. Throughout this section, we consider $q=p^n$ and we let $\tr$ denote the trace function from $\Fq$ to $\Fp$. The following result will be useful to compute the number of solutions of Eq.~\eqref{item300} in the case where $d_1=\dots=d_s=2$.

\begin{lemma}\label{item326}
	For $t|n$, $a\in\F_{q}^*$ and $\lambda\in\Fp$, let $N_\lambda$ denote the number of solutions of $\tr_{q,p}\big(ax^{2}\big)=\lambda$ over $\F_{p^{t}}$ and let $\hat{a}=\tr_{q,p^t}(a)$. Assume that $\hat{a}\neq 0$. Then
	$$N_\lambda=\begin{cases}
	p^{t-1}+ \chi(\hat{a})p^{t/2-1} ,&\text{ if }t\text{ is even}\text{ and }p\equiv 1\pmod{4};\\
	p^{t-1}+ i^t\chi(\hat{a})p^{t/2-1} ,&\text{ if }t\text{ is even}\text{ and }p\equiv 3\pmod{4};\\
	p^{t-1}+\chi(-\lambda \hat{a})p^{(t-1)/2} ,&\text{ if }t\text{ is odd}\text{ and }p\equiv 1\pmod{4},\\
	p^{t-1}+i^{t+1} \chi(-\lambda \hat{a})p^{(t-1)/2} ,&\text{ if }t\text{ is odd}\text{ and }p\equiv 3\pmod{4},\\
	\end{cases}$$
	if $\lambda\neq 0$ and
	$$N_0=\begin{cases}
	p^{t-1}-(p-1)\chi(\hat{a}) p^{t/2-1},&\text{ if }t\text{ is even}\text{ and }p\equiv 1\pmod{4};\\
	p^{t-1}-(p-1) i^t\chi(\hat{a}) p^{t/2-1},&\text{ if }t\text{ is even}\text{ and }p\equiv 3\pmod{4},\\
	p^{t-1},&\text{ if }t\text{ is odd,}
	\end{cases}$$
	where $i$ is the imaginary unity and $\chi$ denotes the quadratic multiplicative character of $\F_{p^t}$.
\end{lemma}

\begin{proof}
	Let $\psi(x)=\exp\left((2\pi i)\tr_{p^t,p}(x)/p\right)$, the canonical additive character, let $\delta\in\F_{p^t}$ be an element such that $\tr_{p^t,p}(\delta)=\lambda$ and let $\hat{a}=\tr_{q,p^t}(a)$. We observe that if $c\in\F_{p^t}$, then $\tr_{q,p}\big(ac^{2}\big)=\tr_{p^t,p}\big(\tr_{q,p^t}(a)c^{2}\big)=\tr_{p^t,p}\big(\hat{a}c^{2}\big)$ so that $\tr_{q,p}\big(ac^{2}\big)=\lambda$ if and only if $\psi(\hat{a}c^2-\delta)=1$. Therefore,
	$$\begin{aligned}pN_\lambda&=
	\sum_{c\in\F_{p^t}}\left[1+\dots+\psi(\hat{a}c^2-\delta)^{p-1}\right]\\
	&=\sum_{c\in\F_{p^t}}\left[1+\dots+\psi(\hat{a}c-\delta)^{p-1}\right]\left[1+\chi(c)\right]\\
	&=p^t+\sum_{\ell=1}^{p-1}\psi(-\delta)^\ell\chi(\ell^{-1} \hat{a}^{-1})\sum_{z\in\F_{p^t}}\psi(z)\chi(z)\\
	&=p^t+\sum_{\ell=1}^{p-1}\psi(-\delta)^\ell\chi(\ell^{-1} \hat{a}^{-1})G(\chi),\\
	\end{aligned}$$
	where $G(\chi)$ is the Gauss sum of $\chi$ over $\F_{p^t}$. Now, we split the proof into two cases:
	\begin{itemize}
		\item If $\lambda= 0$, then
		$$\sum_{\ell=1}^{p-1}\psi(-\delta)^\ell\chi(\ell^{-1} \hat{a}^{-1})=\sum_{\ell=1}^{p-1}\chi(\ell^{-1} \hat{a}^{-1})=\begin{cases}
		(p-1)\chi(\hat{a}),&\text{ if }t\text{ is even;}\\
		0,&\text{ if }t\text{ is odd.}\\
		\end{cases}$$
		\item If $\lambda\neq 0$, then
		$$\begin{aligned}
		\sum_{\ell=1}^{p-1}\psi(-\delta)^\ell\chi(\ell^{-1} \hat{a}^{-1})&=\chi\left(\tfrac{1}{\hat{a}}\right)\sum_{\ell\in\F_p^*}\psi(-\delta)^\ell\overline\chi(\ell)=\chi(\tfrac{1}{\hat{a}})\sum_{\ell\in\F_p^*}\psi(-\delta\ell)\overline\chi(\ell)\\
		&=\chi\left(\tfrac{-\lambda}{\hat{a}}\right)\sum_{\ell\in\F_p^*}\psi(\ell)\overline\chi(\ell)=\chi\left(\tfrac{-\lambda}{\hat{a}}\right)\sum_{\ell\in\F_p^*}\psi(\ell)\chi(\ell)\\
		&=\begin{cases}
		-\chi\left(\hat{a}\right),&\text{ if }t\text{ is even;}\\
		\chi\left(-\lambda \hat{a}\right)G_p(\chi),&\text{ if }t\text{ is odd,}\\
		\end{cases}\\
		\end{aligned}$$
		where $G_p(\chi)$ is the Gauss sum of $\chi$ over $\Fp$. Our result follows directly by using Theorem 5.15 of \cite{Lidl} in the values of $G(\chi)$ and $G_p(\chi)$.
	\end{itemize}
\end{proof}

\begin{proposition}\label{item329}.
	For $c,a_1,\dots,a_s\in\Fq$, let $\hat{a}_j=\tr_{q,p^{t_j}}(ca_j)$ and let $S_{j}(c):=\sum_{x\in\scalebox{0.7}{$\displaystyle \F_{p^{t_j}}^*$}}\psi_{ca_j}\left(x^{2}\right)$. If $\hat{a}_j=0$, then $S_j(c)=p^{t_j}$. If $\hat{a}_j\neq0$, then
	$$S_{j}(c)=\begin{cases}
	-\chi^{(j)}(\hat{a}_j)p^{t_j/2} ,&\text{ if }t_j\text{ is even}\text{ and }p\equiv 1\pmod{4};\\
	- i^{t_j}\chi^{(j)}(\hat{a}_j)p^{t_j/2} ,&\text{ if }t_j\text{ is even}\text{ and }p\equiv 3\pmod{4};\\
	\chi^{(j)}(\hat{a}_j)p^{t_j/2} ,&\text{ if }t_j\text{ is odd}\text{ and }p\equiv 1\pmod{4};\\
	i^{t_j}\chi^{(j)}(\hat{a}_j)p^{t_j/2},&\text{ if }t_j\text{ is odd}\text{ and }p\equiv 3\pmod{4},\\
	\end{cases}$$
	where $i$ denotes the imaginary unity and $\chi^{(j)}$ denotes the quadratic character in $\F_{p^{t_j}}$.
\end{proposition}

\begin{proof}
	 For a fixed $j\in\{1,\dots,s\}$,  we observe that
	\begin{equation}\label{item327}
	S_{j}(c)=\sum_{\lambda\in\Fp} \exp\left(\frac{2\pi i\lambda}{p}\right)N_\lambda,
	\end{equation}
	where $N_\lambda$ is denote the number of solutions of $\tr_{q,p}\big(ax^{2}\big)=\lambda$ over $\F_{p^{t_j}}$. Assume that $t_j$ is odd. If $p\equiv 1\pmod{4}$, then Eq.~\eqref{item327} and Lemma~\ref{item326} imply that
	$$\sum_{\lambda\in\Fp} e^{\frac{2\pi i\lambda}{p}}N_\lambda=\chi^{(j)}(\hat{a}_j)p^{\frac{t_j-1}{2}}\sum_{\lambda\in\Fp^*}e^{\frac{2\pi i\lambda}{p}}\chi^{(j)}(\lambda)=\chi^{(j)}(\hat{a}_j)p^{\frac{t_j-1}{2}}G_p(\chi).$$
	If $p\equiv 3\pmod{4}$, then Eq.~\eqref{item327} and Lemma~\ref{item326} imply that
	$$\sum_{\lambda\in\Fp} e^{\frac{2\pi i\lambda}{p}}N_\lambda=-i^{t_j+1}\chi^{(j)}(\hat{a}_j)p^{\frac{t_j-1}{2}}\sum_{\lambda\in\Fp^*}e^{\frac{2\pi i\lambda}{p}}\chi^{(j)}(\lambda)=-i^{t_j+1}\chi^{(j)}(\hat{a}_j)p^{\frac{t_j-1}{2}}G_p(\chi).$$
	By Theorem 5.15 of \cite{Lidl}, we have that
	$$G_p(\chi)=\begin{cases}
	p^{1/2},&\text{ if }p\equiv 1\pmod{4};\\
	i p^{1/2},&\text{ if }p\equiv 3\pmod{4},\\
	\end{cases}$$
	from where our result follows. The case where $t_j$ is even follows directly from Lemma~\ref{item326}.
\end{proof}

The following result is a straightforward application of Lemma~\ref{item324} and Proposition~\ref{item329}

\begin{theorem}\label{item330}
	Let $a_1,\dots,a_s,b\in\F_q$, where $q=p^n$, and let $i$ denotes the imaginary unity. The number of solutions of the equation
	$$a_1 x_1^{2}+\cdots+a_s x_s^{2}=b$$
	with $x_j\in\F_{p^{t_j}}$ is given by
	$$p^{\tfrac{t_1+\dots+t_s}{2}-n}\sum_{c\in\scalebox{0.7}{$\displaystyle \Fq$}} \psi_c(-b)\prod_{j=1}^{s}\Gamma_j(c),$$
	where $\chi^{(j)}$ denotes the quadratic character in $\F_{p^{t_j}}$ and
	$$\Gamma_j(c)=\begin{cases}
	p^{t_j},&\text{ if }\tr_{q,p^{t_j}}(ca_j)=0;\\
	-(-1)^{t_j}\chi^{(j)}\big(\tr_{q,p^{t_j}}(ca_j)\big),&\text{ if }\tr_{q,p^{t_j}}(ca_j)\neq0\text{ and }p\equiv 1\pmod 4;\\
	-(-1)^{t_j}i^{t_j}\chi^{(j)}\big(\tr_{q,p^{t_j}}(ca_j)\big),&\text{ if }\tr_{q,p^{t_j}}(ca_j)\neq0\text{ and }p\equiv 3\pmod 4.\\
	\end{cases}$$
\end{theorem}

Theorem~\ref{item330} generalizes Theorems 6.26 and 6.27 of \cite{Lidl}.

\section{Final Comments and Open Problems}\label{item335}

This paper provided a counting on the number of solutions of diagonal equations of the form $a_1 x_1^{d_1}+\dots+a_s x_s^{d_s}=b$ with restrict solutions sets satisfying conditions on the exponents. Our counting results extend the main result of \cite{wolfmann1992number} and some results of \cite{oliveira2020diagonal} and \cite{cao2016number}. In the general setting where no conditions are imposed under the exponents, an explicit formula for such number is unknown. Indeed, the problem of counting solutions of diagonal equations in a general setting is still an open problem. In \cite{oliveira2020diagonal}, the author studies the conditions on the exponents of diagonal equations in order to find those whose number of points attains Weil's bound in the standard case where the solution sets are not restrict. From here, we propose the following problems in the restrict solutions sets case.

\begin{problem}\label{item331}
	Can one find a sharp bound by means of simple parameters, such as $d_1,\dots,d_s$ and $q$, for the number of solutions of Equation~\eqref{item300} with $x_i\in\F_{p^{t_i}}$? In this case, what are the conditions for which  diagonal equations attains such bound?
\end{problem}

From this, we have the following conjecture.

\begin{conjecture}
If there exists a bound such as proposed in Problem~\ref{item331} explicitly given in terms of $d_1,\dots,d_s$, then the diagonal equations attaining this bound are those satisfying the conditions of Theorem~\ref{item303}.
\end{conjecture}

In the light of Proposition~\ref{item308}, we wonder if it is possible to replace the monomial $x^{p^r+1}$ by polynomials of the form $xL(x)$ where $L(x)$ is a linearized polynomial in $\Fq[x]$. It is easy to verify that $\Phi_a(x)=\tr_{q,p}(axL(x))$ is a quadratic form. However with the present knowledge of quadratic forms it is not clear
whether one can obtain the dimension of the kernel of the associated symmetric bilinear form. Therefore we have the following general problem.
\begin{problem}
	Can we replace the monomials in Equation~\eqref{item300} by polynomials of the form $xL(x)$ and still obtain simple expressions for the number of solutions of these diagonal equations?
\end{problem}

\section{Acknowledgments}

I would like to thank Lucas Reis for suggesting the problem. This study was financed in part by the Coordena\c{c}\~{a}o de Aperfei\c{c}oamento de Pessoal de N\'{i}vel Superior - Brasil (CAPES) - Finance Code 001.

\printbibliography

\end{document}


Too many Alaskans have been impacted by sexual violence. You are not alone.
After an assault, it may be difficult to know how to feel, if you should tell someone, or what to do next. You may be considering working with the criminal justice system; you may have already reported to the police, and are not sure what the next steps are. There is no right or wrong way to react, and the decision of what steps to take afterwards should be made on what will be best for you. Healing from sexual assault or abuse is a process, and that process looks different for everyone.
If you are unsure whether reporting your assault is the best choice for you, consider speaking to an advocate. An advocate is a trauma-informed professional who has received specialized training to provide support and resources to survivors of sexual assault and other violent crimes. They are free and confidential. See below for additional information about how advocacy can help.
What does the Advocate do?
Assist in identifying and applying for resources such as crime victim compensation, emergency shelter, case management services, financial resources, etc.
Who does an Advocate work for?
In Alaska, advocates work for nonprofit victim service agencies throughout the state. They are independent of the state and the criminal justice system. Their responsibility is to you, not to the criminal justice system. You do not have to report to police or work with the criminal justice system to work with an advocate.
Will the Advocate tell anyone about me or what happened to me?
No, the advocate will not disclose any information regarding what happened to you. Advocates may only release very limited information about you or your needs with your written permission. The advocate is held under strict confidentiality guidelines, and information will not be shared with anyone else, unless it falls under mandated reporting laws. If you are under the age of 18 years of age, or you disclose that something harmful has happened to a minor or vulnerable adult, an advocate is mandated to file a report about the incident under Alaska statute.
I don't want to talk to the police; do I have to?
You do not have to work with the Alaska State Troopers or any other law enforcement agency if you do not wish to. If you have already reported your assault to them, understand that they may continue their investigation without your cooperation.
Will the person who assaulted me be arrested?
A complex set of factors contribute to an arrest warrant being granted and served. Those decisions are made by a team that includes law enforcement and criminal justice professionals on a case-by-case basis, so it is difficult to know what will occur in a specific case. If the assailant is arrested, they may post bail and be released under certain conditions. You can elect to be notified of release by an automatic service.
What if they don't prosecute the person who assaulted me?
The decision to prosecute is a complex one, and is based on evidence available to the prosecutor. The evidence available and the facts of the case are two separate things. A decision not to prosecute does not mean you are not believed! It just means there is not enough independent evidence available to prove to a jury that it happened that way, beyond a reasonable doubt. Regardless of the prosecutor's decision, the most important thing is that you are following your own healing journey, feel supported, informed, and know that whatever happens in a court process (which is out of your hands) is separate from your own trauma recovery process. An advocate can connect you to resources to help you in your healing.
Anchorage: Abused Women's Aid in Crisis, Inc.
Website and toll-free number for updates on an inmate's current location and tentative release date.
The Alaska Native Justice Center supports individuals who have been physically, emotionally or sexually abused by a partner or are seeking assistance with a civil legal issue related to family law such as divorce, separation, child custody, support or a protective order.
Provides free legal services to victims of crime to help them obtain the rights they are guaranteed under the Alaska constitution and statutes.
Visit this website to find the contact information for your local Trooper post.
The Violent Crimes Compensation Board was established to help mitigate financial losses that are the direct result of violent crimes that occur to Alaskans and visitors to Alaska.
Connection to tribal support and resources. Not a 24-hour hotline.
Statewide coalition of advocacy programs; civil legal support, can connect you to local programs that provide direct services. Not a 24-hour hotline.


text:
e - flux Architecture
Post-Internet Cities - Morten Søndergaard - On Para-logic Practices
Post-Internet
Return to Post-Internet Cities
On Para-logic Practices
Martin Kippenberger, Transportable Subway Entrance, Madulain, Switzerland, 2007. Photo: Kamahele.
Post-Internet Cities
In 1993, artist Martin Kippenberger first envisioned Metro-net, a sculptural project that "reacted" to the impact of the internet on urban existence. Consisting of 1:1 entries of metro stations built out of solid, local materials like concrete and wood, Metro-net was installed in remote locations on the Greek island of Syros, Dawnson City West, Canada, Kassel and Venice. Creating a useless, imaginary network of unknown connectivity, Metro-net speaks to not just what a "post-internet city" could be, but also the paradoxes and problematics inherent within such notions of physical/virtual "reality." Fast-forward to 2015 and Ryan S Jeffery & Boaz Levin's film All that is Solid Melts Into Data, which by exploring the design and development of data centers, addresses what Kippenberger envisioned in 1997: a "digital sphere [that] grows larger, demanding an ever-growing physical infrastructure, effecting and shaping our physical landscape."1
Whereas Metro-net is a personal and very direct artistic response to the internet as the metaphor of the information terminals of decentralized globalization, All that is Solid Melts Into Data is an almost clinical dissection of the material effects of data (and by inference, the internet) on the future conditions of the city. It is possible, I will claim in the following, to see both Metro-net and All that is Solid Melts Into Data as evidence of a new kind of practice in the post-internet city; a practice that is skeptical of existing iterations of both the "city" and "technology" and looks at how these two concepts actually materialize in the spheres of post-internet citizens. Underneath the dissent and antagonistic attitudes, practices such as these aim to establishing a new, other logic—a para-logic—of things in the culture of ubiquitous information.
Film still of Ryan S Jeffery & Boaz Levin's film All that is Solid Melts Into Data, 2015.
The city is the venue of citizens. Historically, it is the complex and multi-layered space of events, which frames the city as an ever-changing political and social public sphere.2 "Everything solid melts into thin air" could be claimed to be the pre-internet, pre-digital premise for the public sphere; wars, revolution, ideological battles, totalitarian tendencies, global capitalism and fragile postwar democracies could be said to be the effects of the modern city.3 From these city-effects emerged modern citizenry as the "agents" of public space. Within the last decades however, this position of the city as public sphere is being challenged and transformed beyond recognition by technological inventions such as the internet and the smartphone. As Jean-François Lyotard famously argued in 1979, the period in which the city was the "grand narrative" of modernity is over.4 Both Metro-net and All that is Solid Melts into Data suggest a post-internet city that has melted, if not into the thin air of Marshal Berman (and Karl Marx), then into the information aesthetics of what Mark Weiser famously called "the culture of ubiquitous computing" and what Lyotard referred to simply as "the condition." The post-internet city is, above all, a manifestation of data and infrastructures we may only imagine, and never see or feel.
Technologies in the post-internet city are, according to Wendy Chun, "second nature: autonomous yet intimate, individual yet collective." They have become part of our habits, and "through habits, we have arguably become our technologies: we click, stream, update, capture, upload, share, grind, link, verify, map, save, trash and troll."5 In this way, society itself may be interpreted as a series of practices that aim to establish order. Every order is an expression of an actual structure, and if we are to envision a transition from the established order, we need to abandon the view that only one single order is possible. The question that both Metro-net and All that is Solid Melts into Data raises is how we as inhabitants of the post-internet city may break with our habits and "retake control" of the all-encompassing disorder of globalization and technology. Within both projects there is a strategy of withdrawal. They point towards different solutions—a creative role played both capital, artists and labor alike—and the ever-present possibility, and dialectics, of antagonism and hegemony.
Another demonstration of how the "post-internet condition" affords para-logic practices is the 2002 project In-line by Swedish sound artist Carl Michael von Hausswolff, in which the habitual turn-call sounds of a bank's waiting-line ticket system in Roskilde, Denmark, was changed to newly composed, "unknown," and "weird," sounds. Both the new and the old sounds were integrated into the computer-system of the bank, highlighting what Freidrich Kittler understood to be the general digitalization of information within the simplest of daily routines.6 By introducing small cracks into the apparently "innocent" surface of a bank visit, Hausswolff unveils everyday life to be constituted by a series of "survival spaces," where according to Jacques Attali, "[sound] indicates the limits of a territory and the way to make oneself heard within it, how-to survive by drawing one's sustenance from it."7
Survival Space
Far from suppressing criticism of everyday life, modern technical progress realizes it. This technicity replaces the criticism of life through … those activities which rise above the everyday, by the critique of everyday life from within: the critique… of the real by the possible and of one aspect of life by another.8
Digital media has realized a situation where technology incessantly reinvents the meaning and redefines the limits everyday life. The ubiquity of technology has made the modalities of criticism (and critical thought) become transitory, fugitive and invisible. This does not mean that criticism in the expanded and increasingly digital field of everyday life is extinct or unimportant, but rather that the critical activity of philosophy, art, poetry, and hermeneutics are ubiquitously present, ultimately meaning that cultural artefacts do not carry or imply the same authority they once did. It is no longer possible to presume there to be a reader of a literary work or a visitor to a museum, not to mention one that shares the same norms, values, and habits of the author or curator. A circuit of cultural order has seemed to have been broken, but what—or whom—will take its place?
In "The Author as Producer," Walter Benjamin calls for the author to be address and enact socio-political questions and problems. While he laid the foundations for understanding our culture of ubiquity, we could ask as a thought experiment how Benjamin would look at and see the world we live in today? Whereas Walter Benjamin, writing in 1934, advocates for the "cultural producer" to intervene in the production process in the manner of an engineer, Geoff Cox and Joasia Krysa argue that "social change does not simply result from resistance to the existing set of conditions, but instead from adapting and transforming the technical apparatus itself."9 The phenomenology of the culture of ubiquitous information relies on the paradox of engineering participation—the transformation of an audience into citizens with political and social responsibilities—and the ability to transform technological apparatuses into "cultural" media—to facilitate, in the vernacular of Lefebvre, a generative and meaningful production of space.
The para-logic producer is an emergent type of subject capable of reflecting on the culture of ubiquitous information as a "survival space."10 Their practices highlight and de-stabilize existing methodologies (political, social, creative) and their hidden technological criteria to reveal the constituent elements of the distributed public sphere we all now inhabit.11 In this way, para-logic practices forge a different network of space-production: drawing sustenance from the fact that the public sphere will never be harmonious, it points towards an investigation into alternative orders of the post-internet city and the political organization of its citizens.
See ➝.
Bruno Latour, Reassembling the Social: An Introduction to Actor-Network Theory (Oxford University Press, 2005).
Marshal Berman, All That is Solid Melts into Air: The Experience of Modernity (Simon and Schuster, 1982).
Jean-François Lyotard, The Postmodern Condition (Les Éditions de Minuit, 1979): 25.
Wendy Hui Kyong Chun, Programmed Visions: Software and Memory (Cambridge, MA: MIT Press, 2011): 22–23.
Frederich Kittler, Gramophone, Film, Typewriter (MIT Press, 1987): 102.
Jacques Attali, Noise: The Political Economy of Music (University of Minnesota Press, 1985): 18.
Henri Lefevbre, The Production of Space (Oxford: Blackwell, 1991): xi.
Engineering Culture: On "The Author as (Digital) Producer", eds. Geoff Cox and Joasia Krysa (Autonomedia, 2005)
Ibid., Lyotard: 25
Such practices favor legitimating discourses that explore paradoxes and anomalies, foreground the critique and destabilization of existing methodologies, and create new ones in their place. Paralogy is therefore an approach that favors dynamic tensions and heterogeneity over operativity and consensus. It is the bending of rules, the creation of new rules, and a self-reflexive awareness of the rules that govern its research and culture.
Post-Internet Cities is a collaborative project between e-flux Architecture and MAAT – Museum of Art, Architecture and Technology within the context of the Utopia/Dystopia exhibition and "Post-Internet Cities" conference, produced in association with Institute for Art History, Faculty of Social Sciences and Humanities – Universidade NOVA de Lisboa and Instituto Superior Técnico – Universidade de Lisboa, and supported by MIT Portugal Program and Millennium bcp Foundation.
Urbanism, Technology
Post-Internet, Citizenship, Public Space
Morten Søndergaard is Associate Professor and Curator of Interactive Media Art at Aalborg University, Denmark. He is co-founder and AAU-coordinator of Erasmus Master in Media Arts Cultures, co-founder (with Peter Weibel) of ISACS – International Sound Art Curating Conference Series, and (with Laura Beloff) the upcoming EVA-Copenhagen symposium.
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