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2025-04-24
User8329

Reply with quote Re: Sudoku16: Minlex Forms Very good !! I did think it an increasingly unlikely scenario that a grid has all the 48 row pairs of high rank - and that an automorphic grid with all similar high ranking might be more likely !!! Well found ! coloin Posts: 2532Joined: 05 May 2005Location: Devon Reply with quote More Automorphic Grids by Mathimagics » Fri Apr 02, 2021 2:38 pm .I have found some more high-ranking automorphic grids. These are not H-grids, but they are close relations, and were found while searching for H-grids.Pursuing my "band knitting" analogy, we need a term for the number of distinct minlex bands (horizontal and vertical) in a grid. The vast majority of grids will have 8 different bands, and H-grids by definition only have one, but any number in between is possible.I suggest the term k-ply. So H-grids are 1-ply, and the most common grids (with all different bands) are 8-ply. I also conjecture that low-ply grids are far more likely to be automorphic than normal grids.The first example is 3-ply, it has rank 17371 and 8 automorphisms:Code: Select all+---------+---------+---------+---------+| 0 1 2 3 | 4 5 6 7 | 8 9 A B | C D E F || 4 5 8 C | 2 3 9 D | 7 1 E F | A B 6 0 || 6 9 B F | 1 C E A | 3 5 0 D | 4 2 7 8 || A E 7 D | 0 8 F B | 2 4 6 C | 1 9 5 3 |+---------+---------+---------+---------+| 1 2 0 E | C 6 3 5 | D A B 9 | 8 F 4 7 || 7 3 D A | F E 8 9 | 4 C 1 2 | 5 6 0 B || B C 5 8 | D 0 2 4 | F 6 7 E | 9 3 A 1 || F 6 4 9 | B A 7 1 | 0 3 5 8 | E C D 2 |+---------+---------+---------+---------+| 2 8 E 7 | 5 1 C 3 | 9 F 4 6 | 0 A B D || 3 0 A 6 | E 9 D F | B 8 C 5 | 2 7 1 4 || 9 B 1 5 | 8 2 4 0 | A 7

2025-04-14
User6912

Boost the performance of your NVIDIA graphics card Home Drivers NVIDIA GeForce Game Ready Driver 474.66 WHQL (Windows 7/8 64-bit) Old Versions Browse by CompanyAdobe, Apowersoft, Ashampoo, Autodesk, Avast, Corel, Cyberlink, Google, iMyFone, iTop, Movavi, PassFab, Passper, Stardock, Tenorshare, Wargaming, Wondershare Sponsored October, 31st 2024 - 593 MB - Freeware Review Screenshots Old Versions NVIDIA GeForce Game Ready Driver 474.66 WHQL (Windows 7/8 64-bit) Date released: 31 Oct 2023 (one year ago) NVIDIA GeForce Game Ready Driver 473.81 WHQL (Windows 7/8 64-bit) Date released: 02 Aug 2022 (3 years ago) NVIDIA GeForce Game Ready Driver 472.12 WHQL (Windows 7/8 64-bit) Date released: 21 Sep 2021 (3 years ago) NVIDIA GeForce Game Ready Driver 471.11 WHQL (Windows 7/8 64-bit) Date released: 22 Jun 2021 (4 years ago) NVIDIA GeForce Game Ready Driver 466.77 WHQL (Windows 7/8 64-bit) Date released: 10 Jun 2021 (4 years ago) NVIDIA GeForce Game Ready Driver 466.47 WHQL (Windows 7/8 64-bit) Date released: 17 May 2021 (4 years ago) NVIDIA GeForce Game Ready Driver 466.11 WHQL (Windows 7/8 64-bit) Date released: 14 Apr 2021 (4 years ago) NVIDIA GeForce Game Ready Driver 465.89 WHQL (Windows 7/8 64-bit) Date released: 03 Apr 2021 (4 years ago) NVIDIA GeForce Game Ready Driver 461.92 WHQL (Windows 7/8 64-bit) Date released: 16 Mar 2021 (4 years ago) NVIDIA GeForce Game Ready Driver 461.40 WHQL (Windows 7/8 64-bit) Date released: 27 Jan 2021 (4 years ago) NVIDIA GeForce Game Ready Driver 461.09 WHQL (Windows 7/8 64-bit) Date released: 08 Jan 2021 (4 years ago) NVIDIA GeForce Game Ready Driver 457.09 WHQL (Windows 7/8 64-bit) Date released: 29 Oct 2020 (4 years ago) NVIDIA GeForce Game Ready Driver 456.71 WHQL (Windows 7/8 64-bit) Date released: 08 Oct 2020 (4 years ago) NVIDIA GeForce Game Ready Driver 452.06 WHQL (Windows 7/8 64-bit) Date released:

2025-04-02
User7511

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2025-04-07
User2962

6 0 4 | D E 7 F | B 1 C A || 7 4 1 5 | D 2 3 C | 9 B 6 A | 0 E F 8 || D E 0 B | 9 F A 1 | 4 C 2 8 | 5 7 6 3 || F C A 6 | B 7 8 E | 5 1 3 0 | D 4 2 9 |+---------+---------+---------+---------+| 3 D C A | F 0 5 6 | 1 7 B E | 9 8 4 2 || 8 B 5 7 | A 1 4 9 | 2 3 C 6 | E F D 0 || 9 2 F 1 | 8 E 7 3 | 0 D 5 4 | 6 C A B || E 0 6 4 | C B D 2 | F A 8 9 | 3 5 7 1 |+---------+---------+---------+---------+Grid rank = 17411, automorphisms = 32Congratulations! I could not even imagine that such HB-grids exist. Very interesting! (Please, correct a typo - "34 grid automorphisms" - of course there must be "32" automorphisms.)Well done!Serg Serg 2018 Supporter Posts: 909Joined: 01 June 2010Location: Russia Reply with quote Re: Sudoku16: Minlex Forms by Mathimagics » Sat Apr 03, 2021 5:31 am Thanks, Serg! I have corrected that typo. We already did have one example of a HB-grid from the start, of course, the "most automorphic" grid. And I will look further for "low-rank" HB-grids, as well as continuing the search from the high-rank end. Mathimagics 2017 Supporter Posts: 1926Joined: 27 May 2015Location: Canberra Reply with quote HB-grids by Mathimagics » Sat Apr 03, 2021 1:12 pm .In a HB-grid, every one of the 48 row/col pairs has the same rank. If A and B are both HB-grids, and rank(A) = rank(B), does it then follow that A and B are isomorphic? Remarkably (to me anyway), the answer is "not necessarily" Here we have 3 ED grids, all of which are HB-grids with rank 0:Perhaps this is only possible with rank 0, because of its unique properties ... Mathimagics 2017 Supporter Posts: 1926Joined: 27 May 2015Location: Canberra Reply with quote HB-grids (Complete) by Mathimagics » Thu Apr 08, 2021 4:37 pm .I have improved the "Band Knit" process considerably, and over the past 5 days have used it to test all the 226,142 known H-bands to see if they

2025-03-28
User2552

MacOS Sonoma 14.4 or later 6GB of available storage space for minimum installation or 72GB of storage space for full Sound Library installation Logic Remote Minimum System Requirements iPad: iPadOS 18.0 or later iPhone: iOS 18.0 or later* iPod touch: iOS 18.0 or later* * Smart Controls, Sound Library navigation, Smart Help, full mixer and access to plug-in controls are available only on iPad. 2,114 definable key and MIDI commands 90 recallable screens 200-step undo history, including plug‑in and mixer editing undo Full plug-in latency compensation, also for external audio and MIDI gear 1/3,840-note resolution for MIDI events Audio file resolution up to 32-bit/192kHz Audio I/O resolution up to 24-bit/192kHz Maximum project length greater than six hours at 96kHz; 13 hours at 44.1kHz Professional dithering algorithms (POW-r, Apogee UV22HR) 64-bit summing engine Up to 16 independent voices per part Over 4,000 predefined chord grids Tablature supports up to 16‑string instruments with any tuning Articulation Sets for intuitive key switching between articulations on Logic and third party instruments Score symbols Notes and rests for various lengths Auto-adapting multi-bar rests Key and time signature symbols Clef signs Slurs, and crescendo and decrescendo symbols Various note head types, including percussion styles Accents, bowing marks, fermatas and other symbols that apply to specific notes Phrasing marks Trills and other ornaments, and tremolo symbols Sustain pedal markings Bar lines, repeat signs, and Segno and Coda markings Chord symbols Chord grids and tablature symbols Page and line break symbols Support for Bravura font MusicXML import and export Plug-ins 108 effect plug-ins including Pedalboard, which includes 35 stompboxes 28 software instrument plug-ins 9 MIDI plug-ins 7 instrument and effect plug-ins with support for up to 7.1 surround 14 effect plug-ins with support for up to 7.1.4 surround Customisable instrument-tuning system, including 97 fixed tuning presets, and equal-tempered, customised and Hermode tuning Sound Library 5,953 total patches for audio, auxiliary, software instrument and output tracks 13,552 Apple Loops covering a wide range of genres 1,250 Sampler instruments 110 Quick Sampler instruments 3,556 Alchemy presets 150 Drum Machine Designer kits with more than 3,066 individual kit piece patches 4,529 plug-in settings for 104 plug-ins 1,060 reverb spaces and warped effects for Space Designer 54 groove patterns for Arpeggiator MIDI plug-in 26 Live Loops grids 4 Tutorial projects 161 patterns and templates for Step Sequencer Up to 1,000 stereo audio channel strips Up to 1,000 software instrument channel strips Up

2025-04-06

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