A finiteness result for circulant core complex Hadamard matrices

2019 ◽  
Vol 571 ◽  
pp. 143-153 ◽  
Author(s):  
Remus Nicoara ◽  
Chase Worley
Author(s):  
N. A. Balonin ◽  
M. B. Sergeev ◽  
J. Seberry ◽  
O. I. Sinitsyna

Introduction: The Hadamard conjecture about the existence of Hadamard matrices in all orders multiple of 4, and the Gauss problem about the number of points in a circle are among the most important turning points in the development of mathematics. They both stimulated the development of scientific schools around the world with an immense amount of works. There are substantiations that these scientific problems are deeply connected. The number of Gaussian points (Z3 lattice points) on a spheroid, cone, paraboloid or parabola, along with their location, determines the number and types of Hadamard matrices.Purpose: Specification of the upper and lower bounds for the number of Gaussian points (with odd coordinates) on a spheroid depending on the problem size, in order to specify the Gauss theorem (about the solvability of quadratic problems in triangular numbers by projections onto the Liouville plane) with estimates for the case of Hadamard matrices. Methods: The authors, in addition to their previous ideas about proving the Hadamard conjecture on the base of a one-to-one correspondence between orthogonal matrices and Gaussian points, propose one more way, using the properties of generalized circles on Z3 .Results: It is proved that for a spheroid, the lower bound of all Gaussian points with odd coordinates is equal to the equator radius R, the upper limit of the points located above the equator is equal to the length of this equator L=2πR, and the total number of points is limited to 2L. Due to the spheroid symmetry in the sector with positive coordinates (octant), this gives the values of R/8 and L/4. Thus, the number of Gaussian points with odd coordinates does not exceed the border perimeter and is no less than the relative share of the sector in the total volume of the figure.Practical significance: Hadamard matrices associated with lattice points have a direct practical significance for noise-resistant coding, compression and masking of video information.


2020 ◽  
Vol 27 (33) ◽  
pp. 5530-5542
Author(s):  
Xiaoqing Ye ◽  
Gang Chen ◽  
Jia Jin ◽  
Binzhong Zhang ◽  
Yinda Wang ◽  
...  

Mixed Lineage Leukemia 1 (MLL1), an important member of Histone Methyltransferases (HMT) family, is capable of catalyzing mono-, di-, and trimethylation of Histone 3 lysine 4 (H3K4). The optimal catalytic activity of MLL1 requires the formation of a core complex consisting of MLL1, WDR5, RbBP5, and ASH2L. The Protein-Protein Interaction (PPI) between WDR5 and MLL1 plays an important role in abnormal gene expression during tumorigenesis, and disturbing this interaction may have a potential for the treatment of leukemia harboring MLL1 fusion proteins. In this review, we will summarize recent progress in the development of inhibitors targeting MLL1- WDR5 interaction.


2017 ◽  
Vol 34 (1) ◽  
pp. 45 ◽  
Author(s):  
Elizard González-Becuar ◽  
Efrén Pérez-Segura ◽  
Ricardo Vega-Granillo ◽  
Luigi Solari ◽  
Carlos Manuel González-León ◽  
...  

Plutonic rocks of the Puerta del Sol area, in central Sonora, represent the extension to the south of the El Jaralito batholith, and are part of the footwall of the Sierra Mazatán metamorphic core complex, whose low-angle detachment fault bounds the outcrops of plutonic rocks to the west. Plutons in the area record the magmatic evolution of the Laramide arc and the Oligo-Miocene syn-extensional plutonism in Sonora. The basement of the area is composed by the ca. 1.68 Ga El Palofierral orthogneiss that is part of the Caborca block. The Laramide plutons include the El Gato diorite (71.29 ± 0.45 Ma, U-Pb), the El Pajarito granite (67.9 ± 0.43 Ma, U-Pb), and the Puerta del Sol granodiorite (49.1 ± 0.46 Ma, U-Pb). The younger El Oquimonis granite (41.78 ± 0.32 Ma, U-Pb) is considered part of the scarce magmatism that in Sonora records a transition to the Sierra Madre Occidental magmatic event. The syn-extensional plutons are the El Garambullo gabbro (19.83 ± 0.18 Ma, U-Pb) and the Las Mayitas granodiorite (19.2 ± 1.2 Ma, K-Ar). A migmatitic event that affected the El Palofierral orthogneiss, El Gato diorite, and El Pajarito granite between ca. 68 and 59 Ma might be related to the emplacement of the El Pajarito granite. The plutons are metaluminous to slightly peraluminous, with the exception of El Oquimonis granite, which is a peraluminous two-mica, garnet-bearing granite. They are mostly high-K calc-alkaline with nearly uniform chondrite-normalized REE and primitive-mantle normalized multielemental patterns that are characteristic of continental margin arcs and resemble patterns reported for other Laramide granites of Sonora. The Laramide and syn-extensional plutons also have Sr, Nd and Pb isotopic ratios that plot within the fields reported for Laramide granites emplaced in the Caborca terrane in northwestern and central Sonora. Nevertheless, and despite their geochemical affinity to continental magmatic arcs, the El Garambullo gabbro and Las Mayitas granodiorite are syn-extensional plutons that were emplaced at ca. 20 Ma during development of the Sierra Mazatán metamorphic core complex. The 40Ar/39Ar and K-Ar ages obtained for the El Palofierral orthogneiss, the Puerta del Sol granodiorite, the El Oquimonis granite, and the El Garambullo gabbro range from 26.3 ± 0.6 to 17.4 ± 1.0 Ma and are considered cooling ages associated with the exhumation of the metamorphic core complex.


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