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Register a new userPartial cubes: structures, characterizations, and constructions
كوبونات جزئية: الهياكل والتشخيصات والبناء
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Sergei Ovchinnikov
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تتألف المكعبات الجزئية من شبكات إيزومترية من المكعبات المتعددة الأبعاد. تلعب الهياكل المحددة على الشبكة بواسطة المكعبات النصفية والعلاقات ديوكوفيتش ووينكلر دوراً مهماً في نظرية المكعبات الجزئية. يستخدم هذه الهياكل في الورقة لتشخيص الشبكات الثنائية والمكعبات الجزئية لأي بعد. وتم إنشاء تشخيصات جديدة وإعطاء دلائل جديدة لبعض النتائج المعروفة. يستخدم العمليات المنتج الكارتزي واللصق والتوسع والضغط في الورقة لإنشاء مكعبات جزئية جديدة من القديمة. وبالأخص، يتم حساب الأبعاد الإيزومترية والشبكية للمكعبات الجزئية المحدودة التي تم الحصول عليها بواسطة هذه العمليات.
Partial cubes are isometric subgraphs of hypercubes. Structures on a graph defined by means of semicubes, and Djokovi{c}s and Winklers relations play an important role in the theory of partial cubes. These structures are employed in the paper to characterize bipartite graphs and partial cubes of arbitrary dimension. New characterizations are established and new proofs of some known results are given. The operations of Cartesian product and pasting, and expansion and contraction processes are utilized in the paper to construct new partial cubes from old ones. In particular, the isometric and lattice dimensions of finite partial cubes obtained by means of these operations are calculated.
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Configurations of subspaces like equichordal and equiisoclinic tight fusion frames, which are in some sense optimally spread apart and which also have reconstruction properties emulating those of orthonormal bases, are useful in various applications, such as wireless communications and quantum information theory. In this paper, a new construction of infinite classes of equichordal tight fusion frames built on semiregular divisible difference sets is presented. Sometimes this construction yields an equiisoclinic packing. Each of the constructed fusion frames is shown to have both a flat representation and a sparse representation. Furthermore, integrality conditions which characterize when equichordal and equiisoclinic fusion frames can have orthonormal bases with entries in a subring of the algebraic integers are proven. Keywords: fusion frame, Grassmannian packing, difference sets, simplex bound, equichordal, equiisoclinic
We show that, for every linear ordering of $[2]^n$, there is a large subcube on which the ordering is lexicographic. We use this to deduce that every long sequence contains a long monotone subsequence supported on an affine cube. More generally, we prove an analogous result for linear orderings of $[k]^n$. We show that, for every such ordering, there is a large subcube on which the ordering agrees with one of approximately $frac{(k-1)!}{2(ln 2)^k}$ orderings.
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In this paper, we obtain a number of new infinite families of Hadamard matrices. Our constructions are based on four new constructions of difference families with four or eight blocks. By applying the Wallis-Whiteman array or the Kharaghani array to the difference families constructed, we obtain new Hadamard matrices of order $4(uv+1)$ for $u=2$ and $vin Phi_1cup Phi_2 cup Phi_3 cup Phi_4$; and for $uin {3,5}$ and $vin Phi_1cup Phi_2 cup Phi_3$. Here, $Phi_1={q^2:qequiv 1pmod{4}mbox{ is a prime power}}$, $Phi_2={n^4in mathbb{N}:nequiv 1pmod{2}} cup {9n^4in mathbb{N}:nequiv 1pmod{2}}$, $Phi_3={5}$ and $Phi_4={13,37}$. Moreover, our construction also yields new Hadamard matrices of order $8(uv+1)$ for any $uin Phi_1cup Phi_2$ and $vin Phi_1cup Phi_2 cup Phi_3$.
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