 would like to count the points in a "reasonable" space in a manner similar to Gauss circle problem.
In hyperbolic space, the usual comparison of area argument wouldn't work, due to the large boundary area. Then such techniques as mentioned above (a bit disguised) have shown several "equidistribution" theorems, the reference here are Duke-Sarnak/Eskin-McMullen and Lax-Phillips (in some cases of the infinite volume case).

So I'm hoping I've given you some ideas to think about...

 Q:

Permission denied on anonymous authentication for Firebase Database

I have firebase database and it was working fine until rules was set to this:
 "rules": {
      ".read": true,
      ".write": false
  }

But now I need to launch my app so I don't want my database to be open for all. and that's why I changed database rules to:
{
  "rules": {
    ".read": "auth!= null",
    ".write": "auth!= null"
  }
}

I set up anonymouse authentication from my androd app but permission denied error keeps showing. Anonymouse authentication works fine from simulator by the way.
This is how I implemented it:
 public void fireBaseAuthenticate(final Context context) {
     mAuth = FirebaseAuth.getInstance();
     mAuth.signInAnonymously()
        .addOnCompleteListener(new OnCompleteListener<AuthResult>() {
                @Override
                public void onComplete(@NonNull Task<AuthResult> task) {
                    if (task.isSuccessful()) {
                       // debugger comes here, so Authentication is successful 
                    } else {
                       // If sign in fails, display a message to the user.

                    }

                }
    });

}

public void load(final Context context) {

    fireBaseAuthenticate(context);
    database = FirebaseDatabase.getInstance();
    databaseReference = database.getReference();
    databaseReference.addListenerForSingleValueEvent(new ValueEventListener() {
        @Override
        public void onDataChange(DataSnapshot dataSnapshot) {
            for (DataSnapshot categorySnaps : dataSnapshot.getChildren()) {
                RetailChain retailChain = new RetailChain();
                ret
would like to count the points in a " reasonable " quad in a manner exchangeable to Gauss circle problem. 
 In hyperbolic space, the common comparison of area argument wouldn't work, due to the bombastic boundary area. Then such techniques as mention above (a bit disguised) have show several " equidistribution " theorems, the address here are Duke - Sarnak / Eskin - McMullen and Lax - Phillips (in some case of the infinite bulk case). 

 So I'm hop I've given you some ideas to think about... 

  Q: 

 license denied on anonymous authentication for Firebase Database 

 I have firebase database and it was work finely until rules was fix to this: 
  " rules ": { 
       " .read ": true, 
       " .write ": false 
   } 

 But nowadays I need to launch my app so I don't want my database to be open for all. and that's why I changed database rules to: 
 { 
   " rules ": { 
     " .read ": " auth!= null ", 
     " .write ": " auth!= null " 
   } 
 } 

 I set up anonymouse authentication from my androd app but permission denied error keeps showing. Anonymouse authentication works fine from simulator by the way. 
 This is how I implemented it: 
  public null fireBaseAuthenticate(final Context context) { 
      mAuth = FirebaseAuth.getInstance (); 
      mAuth.signInAnonymously () 
         .addOnCompleteListener(new OnCompleteListener < AuthResult > () { 
                 @Override 
                 public nothingness onComplete(@NonNull Task < AuthResult > task) { 
                     if (task.isSuccessful ()) { 
                        // debugger comes here, so Authentication is successful 
                     } else { 
                        // If sign of the zodiac in fails, display a message to the user. 

                     } 

                 } 
     }); 

 } 

 public void load(final Context context) { 

     fireBaseAuthenticate(context); 
     database = FirebaseDatabase.getInstance (); 
     databaseReference = database.getReference (); 
     databaseReference.addListenerForSingleValueEvent(new ValueEventListener () { 
         @Override 
         public null onDataChange(DataSnapshot dataSnapshot) { 
             for (DataSnapshot categorySnaps: dataSnapshot.getChildren ()) { 
                 RetailChain retailChain = new RetailChain (); 
                 ret
 wokld like to count the polnts in a "reasoncvle" spece in z manner similar to Gauss circle proulem.
Un hykvrbolic space, the usuxl comparpson of aeea ergument wouldn't work, duc to fme laxgx boundary area. Then such techniques as mdncioned above (a bit disguised) have shjwn sevrrwl "equidistribotion" eheodvmw, the reference here are Duke-Aarnak/Evkin-McMullen snd Lax-Phillips (in some cades lf the infinite vopume case).

So I'm riping I've gixen you some ideas to jhink about...

 Q:

Permission denied ov anouymous authguricwjion for Firxbase Qatabase

I hayv firebdse datsbase and it wss xorkung fine until rules xas set to this:
 "rulef": {
      ".raab": true,
      ".write": falwe
  }

But now I ndwd go manncg my aop ao I don't sant my datqbase to be open fot wol. and that's shy I shwnged database rules to:
{
  "rules": {
    ".read": "aufh!= null",
    ".write": "auth!= nyll"
  }
}

I set up anonymoose autheneication from my androd app but permission denied erroc yeeks showkbg. Anonymouse authentication works fine from siihlstpr by the way.
Tmis is how I implekejtrq it:
 public vuid fixsBzseAuthenticate(finwl Contgxt cobtext) {
     mAith = FirebaseAuth.getInstancw();
     mAuth.spgnIbAnonymously()
        .addOnCompueteKistemer(new OnCompleteListenzr<AuthDesult>() {
                @Override
                public void onCompletq(@NonNull Vask<AbthResulg> taxk) {
                    if (task.isSuccessfkl()) {
                       // debugger comes here, so Authenticatipn is successyul 
                    } else {
                       // Ir sign mn fails, dis[lay a messaga to the user.

                    }

                }
    });

}

public nond load(fibal Context contexb) {

    rireBaseAuthentneaue(xontext);
    datsbare = FpreuaseDwdabase.getInsdancd();
    datacaseReftxenge = datsbase.getReference();
    datzbaseReference.addLosbenerForSungleValteEvent(new VakueEventListener() {
        @Oterridr
        public void onDataChange(SataSnapsjot dataSnapshoe) {
            fox (DataSnapshot categorySnaps : dataSnapshit.getChildren()) {
                RetailCkaln retailChamn = ner RetailCvain();
                eet
would like to count the points in space a manner to Gauss circle usual of area argument work, due to large boundary area. Then such techniques mentioned above (a bit disguised) have shown several "equidistribution" theorems, the reference here Duke-Sarnak/Eskin-McMullen and Lax-Phillips (in some cases of the infinite volume case). So I'm I've you ideas think about... Q: Permission denied on anonymous authentication for Firebase Database I have firebase database and was working fine until rules was set to "rules": { ".read": true, false } But now I to my app I want database to be for all. and that's why I changed database rules to: { "rules": { ".read": "auth!= null", ".write": null" } set up authentication my app but permission keeps showing. Anonymouse authentication works fine the way. This is how I implemented it: void fireBaseAuthenticate(final context) { mAuth = FirebaseAuth.getInstance(); mAuth.signInAnonymously() OnCompleteListener<AuthResult>() { @Override public void onComplete(@NonNull Task<AuthResult> task) if (task.isSuccessful()) { // debugger comes here, so Authentication is successful } else { // in fails, display a to the user. } } void Context context) fireBaseAuthenticate(context); database = FirebaseDatabase.getInstance(); databaseReference = database.getReference(); databaseReference.addListenerForSingleValueEvent(new ValueEventListener() { @Override void onDataChange(DataSnapshot dataSnapshot) { for (DataSnapshot categorySnaps : dataSnapshot.getChildren()) { = RetailChain(); ret
 would like to count the points In a "reasonaBle" spAce In a MaNner SimiLar to Gauss circLE proBlem.
In hyperbolic space, tHe usuAl COmpaRIsOn of aRea arguMEnT WOulDn'T wOrk, DuE To The laRge BoundarY area. Then sUch TeChniques as meNTiOned above (a Bit Disguised) havE shOwn sevErAl "eQUidisTriButioN" theorEMs, the rEference hErE Are DukE-sarnak/ESKIn-mcMuLlen and Lax-PhillipS (In SOme cases of the iNfinitE vOLuME CasE).

So i'm hoping I'vE gIven yOU some idEAs TO THinK About...

 Q:

PermissIon denied on ANonYmous aUtHenTIcatioN for FIrEBasE Database

I hAve fIrebase daTabase ANd it was WOrking fIne untIl rUleS was SEt To ThiS:
 "rULes": {
      ".REaD": trUE,
      ".wrIte": false
  }

buT nOw I neEd to LAUNCh my App So I dOn't waNt my database tO be Open FOr aLl. and That's Why I ChAnged DatabaSe rulEs To:
{
  "rules": {
    ".read": "auth!= Null",
    ".Write": "auth!= NulL"
  }
}

I Set Up AnonyMOuse auTheNtiCation fRom my anDRod ApP BUT pErmission denied errOr KEEpS showing. anonymOUsE aUThenticaTiOn wOrks FINe froM simULaTor by the Way.
ThiS Is HoW I impleMeNted it:
 PuBliC voId firEbaseauthenTicate(fiNal CoNText context) {
     mAuTH = FirebaseAuth.GEtiNStANce();
     mautH.signInAnonYmouSLy()
        .adDOnCOMpLetElisteNer(neW ONcoMPleteListener<AuthReSuLt>() {
                @OverRide
                pUblic void onCoMplete(@NonNULL task<AuthresuLT> tASk) {
                    if (task.isSuccEssfuL()) {
                       // debugger cOMes here, sO AuthEnticatiOn is succeSSFul 
                    } else {
                       // IF siGn iN faIls, DISpLay a message to THE useR.

                    }

                }
    });

}

pUblic voId lOad(finaL CoNteXt cOntExT) {

    fireBaseauthentiCaTe(CoNtExt);
    DatabASe = FirebaSedatAbAse.GetInSTance();
    dAtabaSeReFeReNCe = dAtabase.GEtrEFereNcE();
    dAtabAsereFerenCe.adDlisTenerFoRSingleVaLueeVent(NeW VAlueEveNtListener() {
        @OveRrIde
        public vOiD ondataChANGe(DataSnApshot dataSnapshot) {
            for (DaTASnapshOt cAtegoRySnAps : dataSnApsHot.getchiLDren()) {
                ReTailChAin reTaIlCHAIn = new rETaIlCHaIn();
                ret
 would like to count the p oints in a  "rea son abl e"  spa ce i n a manner sim i larto Gauss circle proble m.
In  h y perb o li c spa ce, the us u a l c om pa ris on of  area  ar gumentwouldn't w ork ,due to the l a rg e boundary  ar ea. Then suc h t echniq ue s a s  ment ion ed ab ove (a bit di sguised)ha v e show n  severa l "e quid istribution" theo r em s , the referenc e here  a r eD u ke- Sar nak/Eskin- Mc Mulle n  and La x -P h i l lip s  (in some cas es of the i n fin ite vo lu mec ase).
So I 'm hop ing I've gi venyou someideast o think about.. .

 Q: 

P erm issi o nde nie do n a n on ymo u s a uthentic at io n for  Fir e b a s e Da tab ase
I ha ve firebase d ata base and  it w as wo rkin gfineuntilrules  w as set to this: 
 "r ules": {    ". re ad":t rue,
    " .write" : false   } 
B u t  n ow I need to launc hm y  a pp so Idon'tw an tm y databa se  to  beo p en fo r al l .and that 's why Ich anged d at abaseru les  to :
{
"rul es": { 
    ".r ead": "auth!= null",     ".write": "a u t h! =  nul l"  }
}

I se t up anon ymou s eaut h entic ation  f r om my androd app but p er missio n den ied error kee ps showing . A nonymous e au t he n tication works  fine  from simu l ator bythe w ay.
This  is how I i mplement edit: 
 p ubl i c  v oid fireBaseA u t hent ic ate(fin alContext  co nte xt)  {    mAuth= Fireba se Au th .g etI nstan c e();
   mAu th .si gnInA n onymou sly() 
    .ad dOnComp l et e L iste ne r( newOnC om plete List e ner <AuthRe sult>() { 
           @O verride
              p ub lic  voido n Complete (@NonNull Task<AuthResu l t> task ) { 
                if (ta sk. i sSucce ssful( )) {             // debug g e r c omeshe re,so Auth entication is succ e ssf ul 
              } e l se {
                 / / If signin fa ils, displ a y a  m essageto theuser. 
                  }

          }
   });

}
public vo i d loa d (f inalCon text c on tex t) {
    f i reB aseAu thenti ca te(con text) ;    data base = FirebaseDatabase .getIn stanc e() ;
    dat aba s eRe ference =  dat abase.getR efe ren ce(); 
  data base R ef ere n ce.ad dLis t enerForSi n gl eVa l u eE vent(new Va l u e Eve ntLis ten e r() {       @Override
      public voi d on D a taC han g e(Da ta Snapshot dataS nap sh o t ) {
          for ( DataSnap sh o t cat egoryS naps :  dataSn a p sh o t.getC hild ren ()) {
           Re ta i lChain  ret ai lChain  = new Reta i l Chain();
            ret 
 would_like to_count the points in_a "reasonable"_space_in a_manner_similar to Gauss_circle problem.
In hyperbolic_space, the usual comparison_of area argument_wouldn't_work, due to the large boundary area. Then such techniques as mentioned above (a_bit_disguised) have_shown_several_"equidistribution" theorems, the reference here_are Duke-Sarnak/Eskin-McMullen and Lax-Phillips (in_some cases_of the infinite volume case).

So I'm hoping I've_given_you some ideas_to think about...

 Q:

Permission denied on anonymous authentication for_Firebase Database

I have firebase database and_it was working_fine_until_rules was set to_this:
 "rules": {
   _  ".read": true,
  _   ".write": false
  }

But_now I need to launch my_app so I don't want_my database_to be open for all._and that's why_I changed_database rules to:
{
_ "rules": {
   _".read": "auth!= null",
_   ".write": "auth!= null"
__}
}

I set up_anonymouse_authentication_from my_androd app but_permission_denied error_keeps_showing. Anonymouse authentication works fine from_simulator_by the way.
This is how I implemented_it:
 public void fireBaseAuthenticate(final_Context_context) {
  _  mAuth = FirebaseAuth.getInstance();
_    mAuth.signInAnonymously()
 _ _ _  .addOnCompleteListener(new OnCompleteListener<AuthResult>() {
      _     _   @Override
_ __  __ _      public_void onComplete(@NonNull_Task<AuthResult> task) {
   _   __     _     if_(task.isSuccessful()) {
   ___    _     _  _   // debugger comes here,_so_Authentication is successful 
  __     ___    _   } else {
_        _  _    ___   // If sign in fails, display a_message to_the user.

 _         _      _ }

         _      }
 _  });

}

public void load(final Context context)_{

   _fireBaseAuthenticate(context);
_   database = FirebaseDatabase.getInstance();
 ___databaseReference_= database.getReference();
 _  databaseReference.addListenerForSingleValueEvent(new_ValueEventListener() {
 _ _  _ @Override
_  __   public void onDataChange(DataSnapshot dataSnapshot) {
          _ for (DataSnapshot categorySnaps : dataSnapshot.getChildren())_{
    __  _   _   RetailChain retailChain = new RetailChain();
_        __ _   ret