Germany 2021/22


1. Bundesliga
2. Bundesliga
3. Liga

1. Bundesliga

Final Table:

 1.FC Bayern M�nchen            34  24  5  5  97-37  77  [C]  Champions
 2.Borussia Dortmund            34  22  3  9  85-52  69
 3.Bayer 04 Leverkusen          34  19  7  8  80-47  64
 4.RB Leipzig                   34  17  7 10  72-37  58
 5.1.FC Union Berlin            34  16  9  9  50-44  57
 6.SC Freiburg                  34  15 10  9  58-46  55
 7.1.FC K�ln                    34  14 10 10  52-49  52
 8.1.FSV Mainz 05               34  13  7 14  50-45  46
 9.TSG 1899 Hoffenheim          34  13  7 14  58-60  46
10.Borussia M�nchengladbach     34  12  9 13  54-61  45
11.Eintracht Frankfurt          34  10 12 12  45-49  42
12.VfL Wolfsburg                34  12  6 16  43-54  42
13.VfL Bochum                   34  12  6 16  38-52  42  [P]
14.FC Augsburg                  34  10  8 16  39-56  38
15.VfB Stuttgart                34   7 12 15  41-59  33
- - - - - - - - - - - - - - - - - - - - - - - - - - - -
16.Hertha BSC Berlin            34   9  6 19  37-71  33       Relegation Playoff
-------------------------------------------------------
17.DSC Arminia Bielefeld        34   5 13 16  27-53  28       Relegated
18.SpVgg Greuther F�rth         34   3  9 22  28-82  18  [P]  Relegated

Round 1
[Aug 13]
M�nchengladbach 1-1 Bayern          
[Aug 14]
Wolfsburg       1-0 Bochum          
Union           1-1 Leverkusen      
Augsburg        0-4 Hoffenheim      
Stuttgart       5-1 Greuther F�rth  
Bielefeld       0-0 Freiburg        
Dortmund        5-2 Frankfurt       
[Aug 15]
Mainz           1-0 Leipzig         
K�ln            3-1 Hertha          

Round 2
[Aug 20]
Leipzig         4-0 Stuttgart       
[Aug 21]
Freiburg        2-1 Dortmund        
Bochum          2-0 Mainz           
Hertha          1-2 Wolfsburg       
Frankfurt       0-0 Augsburg        
Greuther F�rth  1-1 Bielefeld       
Leverkusen      4-0 M�nchengladbach 
[Aug 22]
Hoffenheim      2-2 Union           
Bayern          3-2 K�ln            

Round 3
[Aug 27]
Dortmund        3-2 Hoffenheim      
[Aug 28]
K�ln            2-1 Bochum          
Stuttgart       2-3 Freiburg        
Augsburg        1-4 Leverkusen      
Bielefeld       1-1 Frankfurt       
Mainz           3-0 Greuther F�rth  
Bayern          5-0 Hertha          
[Aug 29]
Union           2-1 M�nchengladbach 
Wolfsburg       1-0 Leipzig         

Round 4
[Sep 11]
Greuther F�rth  0-2 Wolfsburg       
Union           0-0 Augsburg        
Hoffenheim      0-2 Mainz           
Leverkusen      3-4 Dortmund        
Freiburg        1-1 K�ln            
Leipzig         1-4 Bayern          
[Sep 12]
Frankfurt       1-1 Stuttgart       
Bochum          1-3 Hertha          
M�nchengladbach 3-1 Bielefeld       

Round 5
[Sep 17]
Hertha          2-1 Greuther F�rth  
[Sep 18]
Augsburg        1-0 M�nchengladbach 
Bielefeld       0-0 Hoffenheim      
Bayern          7-0 Bochum          
Mainz           0-0 Freiburg        
K�ln            1-1 Leipzig         
[Sep 19]
Stuttgart       1-3 Leverkusen      
Dortmund        4-2 Union           
Wolfsburg       1-1 Frankfurt       

Round 6
[Sep 24]
Greuther F�rth  1-3 Bayern          
[Sep 25]
Union           1-0 Bielefeld       
Leverkusen      1-0 Mainz           
Frankfurt       1-1 K�ln            
Leipzig         6-0 Hertha          
Hoffenheim      3-1 Wolfsburg       
M�nchengladbach 1-0 Dortmund        
[Sep 26]
Bochum          0-0 Stuttgart       
Freiburg        3-0 Augsburg        

Round 7
[Oct 1]
K�ln            3-1 Greuther F�rth  
[Oct 2]
Dortmund        2-1 Augsburg        
Hertha          1-2 Freiburg        
Stuttgart       3-1 Hoffenheim      
Wolfsburg       1-3 M�nchengladbach 
Leipzig         3-0 Bochum          
[Oct 3]
Mainz           1-2 Union           
Bayern          1-2 Frankfurt       
Bielefeld       0-4 Leverkusen      

Round 8
[Oct 15]
Hoffenheim      5-0 K�ln            
[Oct 16]
Union           2-0 Wolfsburg       
Dortmund        3-1 Mainz           
Frankfurt       1-2 Hertha          
Freiburg        1-1 Leipzig         
Greuther F�rth  0-1 Bochum          
M�nchengladbach 1-1 Stuttgart       
[Oct 17]
Leverkusen      1-5 Bayern          
Augsburg        1-1 Bielefeld       

Round 9
[Oct 22]
Mainz           4-1 Augsburg        
[Oct 23]
Bielefeld       1-3 Dortmund        
Bayern          4-0 Hoffenheim      
Leipzig         4-1 Greuther F�rth  
Wolfsburg       0-2 Freiburg        
Hertha          1-0 M�nchengladbach 
[Oct 24]
K�ln            2-2 Leverkusen      
Stuttgart       1-1 Union           
Bochum          2-0 Frankfurt       

Round 10
[Oct 29]
Hoffenheim      2-0 Hertha          
[Oct 30]
Union           2-5 Bayern          
Bielefeld       1-2 Mainz           
Leverkusen      0-2 Wolfsburg       
Dortmund        2-0 K�ln            
Freiburg        3-1 Greuther F�rth  
Frankfurt       1-1 Leipzig         
[Oct 31]
Augsburg        4-1 Stuttgart       
M�nchengladbach 2-1 Bochum          

Round 11
[Nov 5]
Mainz           1-1 M�nchengladbach 
[Nov 6]
Bayern          2-1 Freiburg        
Stuttgart       0-1 Bielefeld       
Bochum          2-0 Hoffenheim      
Wolfsburg       1-0 Augsburg        
Leipzig         2-1 Dortmund        
[Nov 7]
Hertha          1-1 Leverkusen      
K�ln            2-2 Union           
Greuther F�rth  1-2 Frankfurt       

Round 12
[Nov 19]
Augsburg        2-1 Bayern          
[Nov 20]
Bielefeld       2-2 Wolfsburg       
Hoffenheim      2-0 Leipzig         
Leverkusen      1-0 Bochum          
Dortmund        2-1 Stuttgart       
M�nchengladbach 4-0 Greuther F�rth  
Union           2-0 Hertha          
[Nov 21]
Freiburg        0-2 Frankfurt       
Mainz           1-1 K�ln            

Round 13
[Nov 26]
Stuttgart       2-1 Mainz           
[Nov 27]
Hertha          1-1 Augsburg        
Greuther F�rth  3-6 Hoffenheim      
Wolfsburg       1-3 Dortmund        
Bochum          2-1 Freiburg        
K�ln            4-1 M�nchengladbach 
Bayern          1-0 Bielefeld       
[Nov 28]
Frankfurt       2-1 Union           
Leipzig         1-3 Leverkusen      

Round 14
[Dec 3]
Union           2-1 Leipzig         
[Dec 4]
Leverkusen      7-1 Greuther F�rth  
Hoffenheim      3-2 Frankfurt       
Augsburg        2-3 Bochum          
Bielefeld       1-1 K�ln            
Mainz           3-0 Wolfsburg       
Dortmund        2-3 Bayern          
[Dec 5]
Stuttgart       2-2 Hertha          
M�nchengladbach 0-6 Freiburg        

Round 15
[Dec 10]
K�ln            0-2 Augsburg        
[Dec 11]
Bayern          2-1 Mainz           
Leipzig         4-1 M�nchengladbach 
Bochum          1-1 Dortmund        
Freiburg        1-2 Hoffenheim      
Hertha          2-0 Bielefeld       
Wolfsburg       0-2 Stuttgart       
[Dec 12]
Greuther F�rth  1-0 Union           
Frankfurt       5-2 Leverkusen      

Round 16
[Dec 14]
Stuttgart       0-5 Bayern          
Mainz           4-0 Hertha          
Wolfsburg       2-3 K�ln            
Bielefeld       2-0 Bochum          
[Dec 15]
M�nchengladbach 2-3 Frankfurt       
Union           0-0 Freiburg        
Dortmund        3-0 Greuther F�rth  
Leverkusen      2-2 Hoffenheim      
Augsburg        1-1 Leipzig         

Round 17
[Dec 17]
Bayern          4-0 Wolfsburg       
[Dec 18]
Frankfurt       1-0 Mainz           
Bochum          0-1 Union           
Hoffenheim      1-1 M�nchengladbach 
Greuther F�rth  0-0 Augsburg        
Leipzig         0-2 Bielefeld       
Hertha          3-2 Dortmund        
[Dec 19]
Freiburg        2-1 Leverkusen      
K�ln            1-0 Stuttgart       

Halfway Table:

 1.FC Bayern M�nchen            17  14  1  2  56-16  43  [C]
 2.Borussia Dortmund            17  11  1  5  41-26  34
 3.SC Freiburg                  17   8  5  4  28-16  29
 4.Bayer 04 Leverkusen          17   8  4  5  40-28  28
 5.TSG 1899 Hoffenheim          17   8  4  5  35-26  28
 6.Eintracht Frankfurt          17   7  6  4  27-24  27
 7.1.FC Union Berlin            17   7  6  4  23-21  27
 8.1.FC K�ln                    17   6  7  4  27-27  25
 9.1.FSV Mainz 05               17   7  3  7  25-17  24
10.RB Leipzig                   17   6  4  7  30-22  22
11.Hertha BSC Berlin            17   6  3  8  20-35  21
12.VfL Bochum                   17   6  2  9  16-26  20  [P]
13.VfL Wolfsburg                17   6  2  9  17-29  20
14.Borussia M�nchengladbach     17   5  4  8  22-32  19
15.FC Augsburg                  17   4  6  7  17-26  18
- - - - - - - - - - - - - - - - - - - - - - - - - - - -
16.VfB Stuttgart                17   4  5  8  22-31  17
-------------------------------------------------------
17.DSC Arminia Bielefeld        17   3  7  7  14-22  16
18.SpVgg Greuther F�rth         17   1  2 14  13-49   5  [P]

Round 18
[Jan 7]
Bayern          1-2 M�nchengladbach 
[Jan 8]
Hoffenheim      3-1 Augsburg        
Leverkusen      2-2 Union           
Greuther F�rth  0-0 Stuttgart       
Leipzig         4-1 Mainz           
Freiburg        2-2 Bielefeld       
Frankfurt       2-3 Dortmund        
[Jan 9]
Hertha          1-3 K�ln            
Bochum          1-0 Wolfsburg       

Round 19
[Jan 14]
Dortmund        5-1 Freiburg        
[Jan 15]
Mainz           1-0 Bochum          
K�ln            0-4 Bayern          
Wolfsburg       0-0 Hertha          
Union           2-1 Hoffenheim      
Stuttgart       0-2 Leipzig         
M�nchengladbach 1-2 Leverkusen      
[Jan 16]
Augsburg        1-1 Frankfurt       
Bielefeld       2-2 Greuther F�rth  

Round 20
[Jan 21]
Frankfurt       0-2 Bielefeld       
[Jan 22]
Greuther F�rth  2-1 Mainz           
Freiburg        2-0 Stuttgart       
Leverkusen      5-1 Augsburg        
M�nchengladbach 1-2 Union           
Hoffenheim      2-3 Dortmund        
Bochum          2-2 K�ln            
[Jan 23]
Leipzig         2-0 Wolfsburg       
Hertha          1-4 Bayern          

Round 21
[Feb 4]
Hertha          1-1 Bochum          
[Feb 5]
Mainz           2-0 Hoffenheim      
Stuttgart       2-3 Frankfurt       
K�ln            1-0 Freiburg        
Augsburg        2-0 Union           
Bielefeld       1-1 M�nchengladbach 
Bayern          3-2 Leipzig         
[Feb 6]
Dortmund        2-5 Leverkusen      
Wolfsburg       4-1 Greuther F�rth  

Round 22
[Feb 11]
Leipzig         3-1 K�ln            
[Feb 12]
M�nchengladbach 3-2 Augsburg        
Frankfurt       0-2 Wolfsburg       
Greuther F�rth  2-1 Hertha          
Bochum          4-2 Bayern          
Freiburg        1-1 Mainz           
Leverkusen      4-2 Stuttgart       
[Feb 13]
Union           0-3 Dortmund        
Hoffenheim      2-0 Bielefeld       

Round 23
[Feb 18]
Mainz           3-2 Leverkusen      
[Feb 19]
Bielefeld       1-0 Union           
Augsburg        1-2 Freiburg        
Stuttgart       1-1 Bochum          
Wolfsburg       1-2 Hoffenheim      
K�ln            1-0 Frankfurt       
[Feb 21]
Bayern          4-1 Greuther F�rth  
Dortmund        6-0 M�nchengladbach 
Hertha          1-6 Leipzig         

Round 24
[Feb 25]
Hoffenheim      2-1 Stuttgart       
[Feb 26]
Union           3-1 Mainz           
Leverkusen      3-0 Bielefeld       
M�nchengladbach 2-2 Wolfsburg       
Freiburg        3-0 Hertha          
Greuther F�rth  1-1 K�ln            
Frankfurt       0-1 Bayern          
[Feb 27]
Bochum          0-1 Leipzig         
Augsburg        1-1 Dortmund        

Round 25
[Mar 4]
Bielefeld       0-1 Augsburg        
[Mar 5]
Bayern          1-1 Leverkusen      
Hertha          1-4 Frankfurt       
Leipzig         1-1 Freiburg        
Bochum          2-1 Greuther F�rth  
Wolfsburg       1-0 Union           
Stuttgart       3-2 M�nchengladbach 
[Mar 6]
K�ln            0-1 Hoffenheim      
[Mar 16]
Mainz           0-1 Dortmund        

Round 26
[Mar 12]
Union           1-1 Stuttgart       
Freiburg        3-2 Wolfsburg       
Hoffenheim      1-1 Bayern          
M�nchengladbach 2-0 Hertha          
[Mar 13]
Leverkusen      0-1 K�ln            
Dortmund        1-0 Bielefeld       
Frankfurt       2-1 Bochum          
Greuther F�rth  1-6 Leipzig         
[Apr 6]
Augsburg        2-1 Mainz           

Round 27
[Mar 18]
Bochum          0-2 M�nchengladbach [abandoned at 0-2 in 70' due to crowd
[Mar 19]                             trouble; result stood]
Mainz           4-0 Bielefeld       
Hertha          3-0 Hoffenheim      
Greuther F�rth  0-0 Freiburg        
Stuttgart       3-2 Augsburg        
Bayern          4-0 Union           
[Mar 20]
Leipzig         0-0 Frankfurt       
Wolfsburg       0-2 Leverkusen      
K�ln            1-1 Dortmund        

Round 28
[Apr 1]
Union           1-0 K�ln            
[Apr 2]
Bielefeld       1-1 Stuttgart       
Leverkusen      2-1 Hertha          
Frankfurt       0-0 Greuther F�rth  
Freiburg        1-4 Bayern          
Hoffenheim      1-2 Bochum          
Dortmund        1-4 Leipzig         
[Apr 3]
Augsburg        3-0 Wolfsburg       
M�nchengladbach 1-1 Mainz           

Round 29
[Apr 8]
Stuttgart       0-2 Dortmund        
[Apr 9]
K�ln            3-2 Mainz           
Bayern          1-0 Augsburg        
Greuther F�rth  0-2 M�nchengladbach 
Wolfsburg       4-0 Bielefeld       
Hertha          1-4 Union           
[Apr 10]
Bochum          0-0 Leverkusen      
Frankfurt       1-2 Freiburg        
Leipzig         3-0 Hoffenheim      

Round 30
[Apr 16]
Mainz           0-0 Stuttgart       
Dortmund        6-1 Wolfsburg       
Augsburg        0-1 Hertha          
Freiburg        3-0 Bochum          
M�nchengladbach 1-3 K�ln            
[Apr 17]
Bielefeld       0-3 Bayern          
Union           2-0 Frankfurt       
Hoffenheim      0-0 Greuther F�rth  
Leverkusen      0-1 Leipzig         

Round 31
[Apr 22]
Wolfsburg       5-0 Mainz           
[Apr 23]
K�ln            3-1 Bielefeld       
Frankfurt       2-2 Hoffenheim      
Leipzig         1-2 Union           
Freiburg        3-3 M�nchengladbach 
Greuther F�rth  1-4 Leverkusen      
Bayern          3-1 Dortmund        
[Apr 24]
Bochum          0-2 Augsburg        
Hertha          2-0 Stuttgart       

Round 32
[Apr 29]
Union           1-1 Greuther F�rth  
[Apr 30]
Mainz           3-1 Bayern          
Bielefeld       1-1 Hertha          
Dortmund        3-4 Bochum          
Augsburg        1-4 K�ln            
Stuttgart       1-1 Wolfsburg       
Hoffenheim      3-4 Freiburg        
[May 2]
Leverkusen      2-0 Frankfurt       
M�nchengladbach 3-1 Leipzig         

Round 33
[May 6]
Bochum          2-1 Bielefeld       
[May 7]
K�ln            0-1 Wolfsburg       
Freiburg        1-4 Union           
Greuther F�rth  1-3 Dortmund        
Hoffenheim      2-4 Leverkusen      
Hertha          1-2 Mainz           
[May 8]
Frankfurt       1-1 M�nchengladbach 
Bayern          2-2 Stuttgart       
Leipzig         4-0 Augsburg        

Round 34 [May 14]
Union           3-2 Bochum          
Mainz           2-2 Frankfurt       
Bielefeld       1-1 Leipzig         
Leverkusen      2-1 Freiburg        
Dortmund        2-1 Hertha          
M�nchengladbach 5-1 Hoffenheim      
Augsburg        2-1 Greuther F�rth  
Stuttgart       2-1 K�ln            
Wolfsburg       2-2 Bayern          

Final Table:

 1.FC Bayern M�nchen            34  24  5  5  97-37  77  [C]  Champions
 2.Borussia Dortmund            34  22  3  9  85-52  69
 3.Bayer 04 Leverkusen          34  19  7  8  80-47  64
 4.RB Leipzig                   34  17  7 10  72-37  58
 5.1.FC Union Berlin            34  16  9  9  50-44  57
 6.SC Freiburg                  34  15 10  9  58-46  55
 7.1.FC K�ln                    34  14 10 10  52-49  52
 8.1.FSV Mainz 05               34  13  7 14  50-45  46
 9.TSG 1899 Hoffenheim          34  13  7 14  58-60  46
10.Borussia M�nchengladbach     34  12  9 13  54-61  45
11.Eintracht Frankfurt          34  10 12 12  45-49  42
12.VfL Wolfsburg                34  12  6 16  43-54  42
13.VfL Bochum                   34  12  6 16  38-52  42  [P]
14.FC Augsburg                  34  10  8 16  39-56  38
15.VfB Stuttgart                34   7 12 15  41-59  33
- - - - - - - - - - - - - - - - - - - - - - - - - - - -
16.Hertha BSC Berlin            34   9  6 19  37-71  33       Relegation Playoff
-------------------------------------------------------
17.DSC Arminia Bielefeld        34   5 13 16  27-53  28       Relegated
18.SpVgg Greuther F�rth         34   3  9 22  28-82  18  [P]  Relegated

Promotion/Relegation Playoff 1./2. Bundesliga

First Leg [May 19]
Hertha          0-1 Hamburg         

Second Leg [May 23]
Hamburg         0-2 Hertha          

NB: both remain at former level

2. Bundesliga

Final Table:

 1.FC Schalke 04 Gelsenkirchen  34  20  5  9  72-44  65  [R]  Promoted
 2.SV Werder Bremen             34  18  9  7  65-43  63  [R]  Promoted
-------------------------------------------------------
 3.Hamburger SV                 34  16 12  6  67-35  60       Promotion Playoff
- - - - - - - - - - - - - - - - - - - - - - - - - - - -
 4.SV Darmstadt 98              34  18  6 10  71-46  60
 5.FC Sankt Pauli               34  16  9  9  61-46  57
 6.1.FC Heidenheim              34  15  7 12  43-45  52
 7.SC Paderborn 07              34  13 12  9  56-44  51
 8.1.FC N�rnberg                34  14  9 11  49-49  51
 9.Holstein Kiel                34  12  9 13  46-54  45
10.Fortuna D�sseldorf           34  11 11 12  45-42  44
11.Hannover 96                  34  11  9 14  35-49  42
12.Karlsruher SC                34   9 14 11  54-55  41
13.FC Hansa Rostock             34  10 11 13  41-52  41  [P]
14.SV Sandhausen                34  10 11 13  42-54  41
15.SSV Jahn Regensburg          34  10 10 14  50-51  40
- - - - - - - - - - - - - - - - - - - - - - - - - - - -
16.Dynamo Dresden               34   7 11 16  33-46  32  [P]  Relegation Playoff
-------------------------------------------------------
17.FC Erzgebirge Aue            34   6  8 20  32-72  26       Relegated
18.FC Ingolstadt 04             34   4  9 21  30-65  21  [P]  Relegated

Promotion/Relegation Playoff 2./3. Bundesliga

First Leg [May 20]
Kaiserslautern  0-0 Dresden         

Second Leg [May 24]
Dresden         0-2 Kaiserslautern  

NB: Kaiserslautern promoted, Dynamo Dresden relegated

3. Liga

Final Table:

 1.1.FC Magdeburg               36  24  6  6  83-39  78       Promoted
 2.Eintracht Braunschweig       36  18 10  8  61-36  64  [R]  Promoted
-------------------------------------------------------
 3.1.FC Kaiserslautern          36  18  9  9  56-27  63       Promotion Playoff
- - - - - - - - - - - - - - - - - - - - - - - - - - - -
 4.TSV 1860 M�nchen             36  17 10  9  67-50  61
 5.SV Waldhof Mannheim          36  16 12  8  58-40  60
 6.VfL Osnabr�ck                36  16 10 10  56-48  58  [R]
 7.1.FC Saarbr�cken             36  14 11 11  50-44  53
 8.SV Wehen Wiesbaden           36  14  9 13  49-44  51
 9.Borussia Dortmund II         36  14  7 15  51-48  49  [P]
10.FSV Zwickau                  36  11 14 11  46-44  47
11.SC Freiburg II               36  12 11 13  34-42  [P]
12.SV Meppen                    36  13  8 15  47-60  47
13.FC Viktoria K�ln             36  12  9 15  39-52  45
14.Hallescher FC                36  10 13 13  46-48  43
15.MSV Duisburg                 36  13  3 20  46-71  42
16.SC Verl                      36  10 10 16  56-66  40
-------------------------------------------------------
17.FC Viktoria 1889 Berlin      36  10  7 19  44-62  37  [P]  Relegated
18.FC W�rzburger Kickers        36   7  9 20  34-59  30  [R]  Relegated
19.TSV Havelse                  36   5  8 23  28-71  23  [P]  Relegated
 -.T�rkg�c� M�nchen             withdrew during the season; all results annulled

Promotion Playoffs for 3. Bundesliga

NB: only the winners of the RL Nordost (BFC Dynamo) and the RL Nord (VfB Oldenburg)
    enter the playoff; the winners of the RL Bayern (SpVgg Bayreuth), the RL West
    (Rot-Weiss Essen) and the RL S�dwest (SV 07 Elversberg) are promoted directly

First Leg [May 28]
Dynamo Berlin   0-2 Oldenburg       

Second Leg [Jun 4]
Oldenburg       1-2 Dynamo Berlin   

NB: VfB Oldenburg promoted

2020/21

2022/23

list of champions

all-time table

list of 2nd level champions

list of amateur champions

oberliga berlin 1945-1963

oberliga nord 1945-1963

oberliga süd 1945-1963

oberliga südwest 1945-1963

oberliga südwest süd 1945-1950

oberliga west 1945-1963

regionalliga nord 63-74

regionalliga süd 63-74

regionalliga west 63-74

list of cup finals

list of super and league cup finals

list of topscorers

list of 2nd level topscorers

list of foundation dates


About this document

Thanks to Martin Arenlind

Prepared and maintained by Karel Stokkermans for the Rec.Sport.Soccer Statistics Foundation

Author: Karel Stokkermans
Last updated: 25 Sep 2022

(C) Copyright Karel Stokkermans and RSSSF 2021/22
You are free to copy this document in whole or part provided that proper acknowledgement is given to the author. All rights reserved.