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βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Ό
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆ
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆ
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠ
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆ
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆ
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“β‰€
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“β‰€β†§
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“β‰€β†§β†΄
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“β‰€β†§β†΄β†₯
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“β‰€β†§β†΄β†₯∏
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“β‰€β†§β†΄β†₯βˆβ‰Ί
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“β‰€β†§β†΄β†₯βˆβ‰Ίβˆ’
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“β‰€β†§β†΄β†₯βˆβ‰Ίβˆ’β‰Ό
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“β‰€β†§β†΄β†₯βˆβ‰Ίβˆ’β‰Όβ†²
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“β‰€β†§β†΄β†₯βˆβ‰Ίβˆ’β‰Όβ†²βŠ³
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“β‰€β†§β†΄β†₯βˆβ‰Ίβˆ’β‰Όβ†²βŠ³βˆ–
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“β‰€β†§β†΄β†₯βˆβ‰Ίβˆ’β‰Όβ†²βŠ³βˆ–β‡
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“β‰€β†§β†΄β†₯βˆβ‰Ίβˆ’β‰Όβ†²βŠ³βˆ–β‡β‰€
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“β‰€β†§β†΄β†₯βˆβ‰Ίβˆ’β‰Όβ†²βŠ³βˆ–β‡β‰€β†§
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“β‰€β†§β†΄β†₯βˆβ‰Ίβˆ’β‰Όβ†²βŠ³βˆ–β‡β‰€β†§β†΄
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“β‰€β†§β†΄β†₯βˆβ‰Ίβˆ’β‰Όβ†²βŠ³βˆ–β‡β‰€β†§β†΄β†•
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“β‰€β†§β†΄β†₯βˆβ‰Ίβˆ’β‰Όβ†²βŠ³βˆ–β‡β‰€β†§β†΄β†•β‹…
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“β‰€β†§β†΄β†₯βˆβ‰Ίβˆ’β‰Όβ†²βŠ³βˆ–β‡β‰€β†§β†΄β†•β‹…β‰Ό
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“β‰€β†§β†΄β†₯βˆβ‰Ίβˆ’β‰Όβ†²βŠ³βˆ–β‡β‰€β†§β†΄β†•β‹…β‰Όβ†²
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“β‰€β†§β†΄β†₯βˆβ‰Ίβˆ’β‰Όβ†²βŠ³βˆ–β‡β‰€β†§β†΄β†•β‹…β‰Όβ†²βˆͺ
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“β‰€β†§β†΄β†₯βˆβ‰Ίβˆ’β‰Όβ†²βŠ³βˆ–β‡β‰€β†§β†΄β†•β‹…β‰Όβ†²βˆͺ∣
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“β‰€β†§β†΄β†₯βˆβ‰Ίβˆ’β‰Όβ†²βŠ³βˆ–β‡β‰€β†§β†΄β†•β‹…β‰Όβ†²βˆͺβˆ£β†½
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“β‰€β†§β†΄β†₯βˆβ‰Ίβˆ’β‰Όβ†²βŠ³βˆ–β‡β‰€β†§β†΄β†•β‹…β‰Όβ†²βˆͺβˆ£β†½β‰»
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“β‰€β†§β†΄β†₯βˆβ‰Ίβˆ’β‰Όβ†²βŠ³βˆ–β‡β‰€β†§β†΄β†•β‹…β‰Όβ†²βˆͺβˆ£β†½β‰»β†₯
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“β‰€β†§β†΄β†₯βˆβ‰Ίβˆ’β‰Όβ†²βŠ³βˆ–β‡β‰€β†§β†΄β†•β‹…β‰Όβ†²βˆͺβˆ£β†½β‰»β†₯≀
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“β‰€β†§β†΄β†₯βˆβ‰Ίβˆ’β‰Όβ†²βŠ³βˆ–β‡β‰€β†§β†΄β†•β‹…β‰Όβ†²βˆͺβˆ£β†½β‰»β†₯≀↧
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“β‰€β†§β†΄β†₯βˆβ‰Ίβˆ’β‰Όβ†²βŠ³βˆ–β‡β‰€β†§β†΄β†•β‹…β‰Όβ†²βˆͺβˆ£β†½β‰»β†₯≀↧↴
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“β‰€β†§β†΄β†₯βˆβ‰Ίβˆ’β‰Όβ†²βŠ³βˆ–β‡β‰€β†§β†΄β†•β‹…β‰Όβ†²βˆͺβˆ£β†½β‰»β†₯≀↧↴↕
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“β‰€β†§β†΄β†₯βˆβ‰Ίβˆ’β‰Όβ†²βŠ³βˆ–β‡β‰€β†§β†΄β†•β‹…β‰Όβ†²βˆͺβˆ£β†½β‰»β†₯β‰€β†§β†΄β†•βˆ
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“β‰€β†§β†΄β†₯βˆβ‰Ίβˆ’β‰Όβ†²βŠ³βˆ–β‡β‰€β†§β†΄β†•β‹…β‰Όβ†²βˆͺβˆ£β†½β‰»β†₯β‰€β†§β†΄β†•βˆβ‰Ό
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“β‰€β†§β†΄β†₯βˆβ‰Ίβˆ’β‰Όβ†²βŠ³βˆ–β‡β‰€β†§β†΄β†•β‹…β‰Όβ†²βˆͺβˆ£β†½β‰»β†₯β‰€β†§β†΄β†•βˆβ‰Όβ†²
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“β‰€β†§β†΄β†₯βˆβ‰Ίβˆ’β‰Όβ†²βŠ³βˆ–β‡β‰€β†§β†΄β†•β‹…β‰Όβ†²βˆͺβˆ£β†½β‰»β†₯β‰€β†§β†΄β†•βˆβ‰Όβ†²β‰Ί
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“β‰€β†§β†΄β†₯βˆβ‰Ίβˆ’β‰Όβ†²βŠ³βˆ–β‡β‰€β†§β†΄β†•β‹…β‰Όβ†²βˆͺβˆ£β†½β‰»β†₯β‰€β†§β†΄β†•βˆβ‰Όβ†²β‰Ίβ‡‚
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“β‰€β†§β†΄β†₯βˆβ‰Ίβˆ’β‰Όβ†²βŠ³βˆ–β‡β‰€β†§β†΄β†•β‹…β‰Όβ†²βˆͺβˆ£β†½β‰»β†₯β‰€β†§β†΄β†•βˆβ‰Όβ†²β‰Ίβ‡‚β†“
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“β‰€β†§β†΄β†₯βˆβ‰Ίβˆ’β‰Όβ†²βŠ³βˆ–β‡β‰€β†§β†΄β†•β‹…β‰Όβ†²βˆͺβˆ£β†½β‰»β†₯β‰€β†§β†΄β†•βˆβ‰Όβ†²β‰Ίβ‡‚β†“β‹Ÿ
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“β‰€β†§β†΄β†₯βˆβ‰Ίβˆ’β‰Όβ†²βŠ³βˆ–β‡β‰€β†§β†΄β†•β‹…β‰Όβ†²βˆͺβˆ£β†½β‰»β†₯β‰€β†§β†΄β†•βˆβ‰Όβ†²β‰Ίβ‡‚β†“β‹Ÿβˆͺ
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“β‰€β†§β†΄β†₯βˆβ‰Ίβˆ’β‰Όβ†²βŠ³βˆ–β‡β‰€β†§β†΄β†•β‹…β‰Όβ†²βˆͺβˆ£β†½β‰»β†₯β‰€β†§β†΄β†•βˆβ‰Όβ†²β‰Ίβ‡‚β†“β‹Ÿβˆͺ↕
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“β‰€β†§β†΄β†₯βˆβ‰Ίβˆ’β‰Όβ†²βŠ³βˆ–β‡β‰€β†§β†΄β†•β‹…β‰Όβ†²βˆͺβˆ£β†½β‰»β†₯β‰€β†§β†΄β†•βˆβ‰Όβ†²β‰Ίβ‡‚β†“β‹Ÿβˆͺ↕⋅
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“β‰€β†§β†΄β†₯βˆβ‰Ίβˆ’β‰Όβ†²βŠ³βˆ–β‡β‰€β†§β†΄β†•β‹…β‰Όβ†²βˆͺβˆ£β†½β‰»β†₯β‰€β†§β†΄β†•βˆβ‰Όβ†²β‰Ίβ‡‚β†“β‹Ÿβˆͺβ†•β‹…βŠ”
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“β‰€β†§β†΄β†₯βˆβ‰Ίβˆ’β‰Όβ†²βŠ³βˆ–β‡β‰€β†§β†΄β†•β‹…β‰Όβ†²βˆͺβˆ£β†½β‰»β†₯β‰€β†§β†΄β†•βˆβ‰Όβ†²β‰Ίβ‡‚β†“β‹Ÿβˆͺβ†•β‹…βŠ”βˆ
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
βˆͺ↕⋅↕⋅≼↲β†₯β‰€β†§β†΄βˆβˆͺβˆβˆ£β†½β‰»β‰Όβ†²β†₯βˆ–βˆ–β†½β‰Όβ†²β‰Όβ†²βˆβŠ”βˆβ†²β†₯βˆ–βˆ–β†½βˆͺβ‡€βˆ–β†•β‰€β†§β†΄βˆ™β†Ώβ‰Όβ‰»β†•β‹…β‰€β†§β†΄βˆβ‰€β†§β†΄βˆ’β†₯βˆ‘βˆ–β†½β†•β†½β‰»β‰Όβ†²β†•β†½β‰»βˆβ‰€β†§β†ͺβˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡ β‰€β†§β†΄βˆβ‰Ίβ‡‚β‡€βˆͺβŠ“βˆΆβŠ”βˆβ†²β†•β†½β‰»βˆβ‰€β†§βŠ”βˆ–βŠ½β‰Όβ†²βŠ³βˆ–β‡βˆ£β†½βˆŽβ‰€β†§β†΄βˆβ‰Ίβ‡‚β†½β‰³β†Ύβ‰€β†§β†΄β†•β†½β‰»β†•β†½β‰»β†•β‹…βˆͺβ‰€β†§β†΄β‰Ίβˆ’βˆβˆ β‰Όβ†²β†•β‹…βˆͺβˆ–βˆ–βŠ½β†₯βŠ”β†₯β†•βˆβ‰Ίβˆ’β†•β‹…β‰Όβ†²β‰€β†§β†ͺβˆ–βŠ½β†•βˆβ‰Έβ‰Ÿβ‡€βˆͺβŠ“β‹…β†Όβ‰šβŠ³βˆ–β‡β‰€β†§β†΄βˆβ‰Όβ†²β‡€β†Έβ‰€β†§β†΄βˆβ†“β†•β†½β‰»β†₯β‰Όβ†²β‹…βˆ–βˆ–β†½β‰Όβ†²βˆ–βˆ–β†½β†•β‹…βˆβ‰Όβ†² βŠ”βˆβ†²βˆβˆβˆβˆβ†²β†•β‹…β†•β‰Ίβˆ’β‰€β†§β†΄βˆβˆ‘βˆ–β‡β‰Όβ†²βŠ³βˆ–β‡βˆβˆβ†“β‰€β†§β†΄βˆ©β†²β‰Ίβ‡‚βˆβ†“β‰€β†§βŠ”βˆ–βŠ½βŠ³βˆ–β‡βˆβ†“β‰€β†§β†΄βˆβ‹…β†•β‰Ίβˆ’β‰Όβ†²βŠ³βˆ–βŠ½β†“β‰―βˆͺβ†•β‹…βŠ”βˆβ†²β†•β†½β‰»β‰€β†§β†΄βŠ”β‰Όβ†²β†•β‹…βˆβˆβŠβ‹…βŠ”βˆβ†²βˆβ†“β‰€β†§β†΄β†₯βˆβ‰Ίβˆ’β‰Όβ†²βŠ³βˆ–β‡β‰€β†§β†΄β†•β‹…β‰Όβ†²βˆͺβˆ£β†½β‰»β†₯β‰€β†§β†΄β†•βˆβ‰Όβ†²β‰Ίβ‡‚β†“β‹Ÿβˆͺβ†•β‹…βŠ”βˆβ†²
As an example, wewrite the numerically estimated mass matrices for thepattern $B_5$ , the matrices are obtainedfor the
emission.
emission.
The bulk kinetic energv of the accelerated electrons. however. is carried to the chromosphere where it is deposited by electron-ion collisions.
The bulk kinetic energy of the accelerated electrons, however, is carried to the chromosphere where it is deposited by electron-ion collisions.
The collision of the beam with (he dense plasma reveals itself bv non-thermal hard X-ray radiation (INR. (vpically between 10β€”100 keV) that. however. constitutes only a small fraction (~10.7) of the total enerey loss.
The collision of the beam with the dense plasma reveals itself by non-thermal hard X-ray radiation (HXR, typically between $-$ 100 keV) that, however, constitutes only a small fraction $\sim 10^{-5}$ ) of the total energy loss.
The bulk energv is translormed into heat. producing an overpressure in the chromosphere as the gas cannot radiate away the energy influx sufficiently rapidly.
The bulk energy is transformed into heat, producing an overpressure in the chromosphere as the gas cannot radiate away the energy influx sufficiently rapidly.
As a consequence. the gas evaporales explosively into the corona as a c10* Ix plasma visible in X-rays curing the eracual phase of a solar flare (Dennis19883).
As a consequence, the gas evaporates explosively into the corona as a $\sim 10^7$ K plasma visible in X-rays during the gradual phase of a solar flare \citep{dennis88}.
. The observed evrosyichrotron radio Iuminosity Ly and the hard X-ray. luminosity Zjsgi are. to first order. proportional to the instantaneous nunmber of [ast electrons aud (therefore to the power E injected into the svstem. while the slowly variable soft N-rav. Iuminositv. Ly is roughly proportional to the accumulated. total enerev Lf in the hot coronal flare plasma.
The observed gyrosynchrotron radio luminosity $L_{\mathrm{R}}$ and the hard X-ray luminosity $L_{\mathrm{HXR}}$ are, to first order, proportional to the instantaneous number of fast electrons and therefore to the power $\dot{E}$ injected into the system, while the slowly variable soft X-ray luminosity $L_{\mathrm{X}}$ is roughly proportional to the accumulated total energy $E$ in the hot coronal flare plasma.
One therelore expects that a relation that is commonly known as (he Neupert Effect! (Neupert1968:DennisZarrvo 1993).
One therefore expects that a relation that is commonly known as the `Neupert Effect' \citep{neupert68,dennis93}.
. Although significant deviations from this scenario have been observed in solar flares (e.g.. heating starting before any hard N-ravs can be detected. or absence of one of the enission (pes discussed above). Chere is strong support for several features of this model in the majority. of solar flares (Dennis&Zarro1993).
Although significant deviations from this scenario have been observed in solar flares (e.g., heating starting before any hard X-rays can be detected, or absence of one of the emission types discussed above), there is strong support for several features of this model in the majority of solar flares \citep{dennis93}.
. For example. (he coincidence to within a fraction of a second of ΗΝ brightenings al a pair of magnetic loop footpoints that are separated by ~10? em requires non-thermal particle velocities (Sakao1994).
For example, the coincidence to within a fraction of a second of HXR brightenings at a pair of magnetic loop footpoints that are separated by $\sim 10^9$ cm requires non-thermal particle velocities \citep{sakao94}.
. It is thought that the same mechanisms should operate in stellar flares. although flares on some classes of stars deviate considerably from the proposed solar analogy.
It is thought that the same mechanisms should operate in stellar flares, although flares on some classes of stars deviate considerably from the proposed solar analogy.
In particular. ejant [lares on RS CVn-tvpe binaries may require mechanisms unknown on the Sun.
In particular, giant flares on RS CVn-type binaries may require mechanisms unknown on the Sun.
RS CVu binaries. commonly consisting of a giant or subelant primary with a main-sequence or subgiant companion in a close orbit. are sources of huninous radio and X-ray emission (Drake.Simon.&Linskv1989).
RS CVn binaries, commonly consisting of a giant or subgiant primary with a main-sequence or subgiant companion in a close orbit, are sources of luminous radio and X-ray emission \citep*{drake89}.
. Very long flare time scales and radio source sizes of order of the intrabinary distance have been modeled in terms of giant. dipole-like magnetospheric structures (Morris.Mitel.&Su1990:Jonesetal.1994) into which high-enerey particles are injected [rom a flare site (Muteletal.1935).. and where Chev lose most of their energy by raciation.
Very long flare time scales and radio source sizes of order of the intrabinary distance have been modeled in terms of giant dipole-like magnetospheric structures \citep*{morris90, jones94} into which high-energy particles are injected from a flare site \citep{mutel85}, and where they lose most of their energy by radiation.
Coordinated observations in A-ravs and radio are required to study (he importance of high-energy electrons in (he heating mechanism.
Coordinated observations in X-rays and radio are required to study the importance of high-energy electrons in the heating mechanism.
Lawleyetal.(1995). reported a Neupert elfect-like behavior during a Large flare on the dMe star AD Leo. where optical (U band) and EUV emissions were used as proxies lor the radiation [rom high-energv electrons (racio. JINR) and from the thermal plasma. respectively,
\citet{hawley95} reported a Neupert effect-like behavior during a large flare on the dMe star AD Leo, where optical (U band) and EUV emissions were used as proxies for the radiation from high-energy electrons (radio, HXR) and from the thermal plasma, respectively.
GΓΌdeletal.(1996). discussed (he first stellar Neupert effect seen in the radio and X-ray bands on the dM35.5e binary UV. Cet. finding similar timine and similar energv budgets as in solar gradual ("tvpe C) events.
\citet{guedel96} discussed the first stellar Neupert effect seen in the radio and X-ray bands on the dM5.5e binary UV Cet, finding similar timing and similar energy budgets as in solar gradual (β€œtype C”) events.
and the torquing terms The torquing terms are (he most difficult. to. integrate nmnunericallv.
and the torquing terms The torquing terms are the most difficult to integrate numerically.
This is because while the terms themselves are at O(e). thev involve quotients of particularly high powers.
This is because while the terms themselves are at $\O\left(\epsilon^4\right)$, they involve quotients of particularly high powers.
Taking derivatives of H' further bloats these bottom and top heavy fractions. and provokes roundolfft errors.
Taking derivatives of $\H^\torq$ further bloats these bottom and top heavy fractions, and provokes roundoff errors.
We argue below (hat (he torquing terms are in anv case uninportant for the known Galactic-center stars.
We argue below that the torquing terms are in any case unimportant for the known Galactic-center stars.
ILence we omit these terms in our numerical work.
Hence we omit these terms in our numerical work.
We can infer the sealing with orbital size @ of the redshift contributions Az of the various perturbative terms in 77,4, with the following deliberation,
We can infer the scaling with orbital size $a$ of the redshift contributions $\Delta z$ of the various perturbative terms in $\H_\star$ with the following deliberation.
Consider a perturbative term Since 7144,star is constant alongnl the orbit. any variation in NH...star has to be balanced by a variation in the unperturbed Hamiltonian.
Consider a perturbative term Since $\H_\star$ is constant along the orbit, any variation in $\Delta\H_\star$ has to be balanced by a variation in the unperturbed Hamiltonian.
Since the latter scales as Haar71/Β«. we have N74,~Ad /a?. giving since the orbital velocity scales as e ~aτιwe have Av hence the redshilt signal ΞΊ A similar argument can be made for the perturbative terms N24.
Since the latter scales as $\H_\star \sim 1/a $, we have $\Delta \H_{star} \sim \Delta a/a^2$ , giving Furthermore, since the orbital velocity scales as $v \sim a^{-1/2}$, we have $\Delta v \sim a^{-3/2}\,\Delta a$ and hence the redshift signal A similar argument can be made for the perturbative terms $\Delta\H^\photon$.
In this case we compare photons emitted from the same point. only with different Hamiltonians.
In this case we compare photons emitted from the same point, only with different Hamiltonians.
While (he redshift signal from (he previous case necessitated ourconsideration of the
While the redshift signal from the previous case necessitated ourconsideration of the
The results of this analvsis are given in Table 4..
The results of this analysis are given in Table \ref{table-jnu}.
Belore we begin the discussion of the results. some words about the normalization values listed in Table 4 are in order.
Before we begin the discussion of the results, some words about the normalization values listed in Table \ref{table-jnu} are in order.
In (he BDO methodfor measuring J(z4). lines are binned in ως) and compared to the ionization model given by Equ. 3..
In the BDO methodfor measuring $J(\nu_{0})$, lines are binned in $\omega(z)$ and compared to the ionization model given by Equ. \ref{eq:dndx},
for an assumed value of 5Β».
for an assumed value of $\beta$.
In (his case. the normalization listed in Table 4 is the parameter in Equ. 1..
In this case, the normalization listed in Table \ref{table-jnu} is the parameter in Equ. \ref{eq:dndzdw},
found from the number of lines in the sample and the maximum likelihood value of 5: where A’ is the total number of lines observed with rest equivalent width greater than Wu. Che limiting equivalent width of the line sample.
found from the number of lines in the sample and the maximum likelihood value of $\gamma$: where ${\cal N}$ is the total number of lines observed with rest equivalent width greater than $W_{\rm lim}$, the limiting equivalent width of the line sample.
For the maximun likelihood solutions for J(rg). we convert line equivalent widths to column densities using the Ly-a curve of erowth and an assumed value of 5. the characteristic Doppler parameter of the lines.
For the maximum likelihood solutions for $J(\nu_{0})$, we convert line equivalent widths to column densities using the $\alpha$ curve of growth and an assumed value of $b$, the characteristic Doppler parameter of the lines.
As we will demonstrate. different. values of 9 and 5 have only a small effect on the value of (24) founcl.
As we will demonstrate, different values of $\beta$ and $b$ have only a small effect on the value of $J(\nu_{0})$ found.
The normalization is given by where .N7.(2) is the limiting column density across each QSO spectrum corresponding to a limiting equivalent width.
The normalization is given by where $N_{\rm min}^{q}(z)$ is the limiting column density across each QSO spectrum corresponding to a limiting equivalent width.
This quantity can be held constant. as in the BDO method. or it can be allowed to vary across each QSO spectrum.
This quantity can be held constant, as in the BDO method, or it can be allowed to vary across each QSO spectrum.
I both of these formulations for the normalization. a proximitv region around the QSO is neglected aud (hat proximity region is either defined bv a velocity cut. eg.
In both of these formulations for the normalization, a proximity region around the QSO is neglected and that proximity region is either defined by a velocity cut, eg.
tq, - 3000 km |. or bv a cut in Ξ±Οƒ) eg. w(t)=0.2.
$z_{\rm em}$ - 3000 km $^{-1}$, or by a cut in $\omega(z)$, eg. $\omega(z)=0.2$.
We also use the standard BDO method to find logL/(4,)]=β€”22.04fi and β€”22.06.$5 for equivalent width thresholds of 0.32 and 0.24 respectively.
We also use the standard BDO method to find $\log[J(\nu_{0})]= -22.04^{+0.43}_{-1.11}$ and $-22.06^{+0.05}_{-0.62}$ for equivalent width thresholds of 0.32 and 0.24 respectively.
Figures. 7((a) and (d) illustrate the 4? of the binned data compared to the BDO ionization model as a function of assumed (24) for these two thresholds.
Figures \ref{fig:chi2}( (a) and (d) illustrate the $\chi^{2}$ of the binned data compared to the BDO ionization model as a function of assumed $J(\nu_{0}$ ) for these two thresholds.
The BDO ionizalion model is expressed in terms of the number of lines per coevolving coordinate: where X.=f(12-zy dz.
The BDO ionization model is expressed in terms of the number of lines per coevolving coordinate: where $X_\gamma= \int (1+z)^{\gamma} dz$ .
This V? curve is very broad. which is reflected in the large error bars and indicates the difficulty in isolating the optimal mean intensity of a weak background
This $\chi^{2}$ curve is very broad, which is reflected in the large error bars and indicates the difficulty in isolating the optimal mean intensity of a weak background
template.
template.
The accuracy of such displacement measurement stronely depends on the magnitude at the. peak and. in the case of our events is at. a. level of LOO mas (Iensed images of the source are usually separated. by less than 1 mas hence only centroid shift can be measured with such accuracy).
The accuracy of such displacement measurement strongly depends on the magnitude at the peak and in the case of our events is at a level of 100 mas (lensed images of the source are usually separated by less than 1 mas hence only centroid shift can be measured with such accuracy).
Because stars randomly blended: together are usually displaced slightly from each other within the seeing clisk. astrometric detection of a shift of the light barvcentre orovides an additional proof for β€œnatural” random blending.
Because stars randomly blended together are usually displaced slightly from each other within the seeing disk, astrometric detection of a shift of the light barycentre provides an additional proof for β€œnatural” random blending.
In the case of the two events presented here there was clearly no shift visible in astrometry of the residuals on the DIA subtracted images.
In the case of the two events presented here there was clearly no shift visible in astrometry of the residuals on the DIA subtracted images.