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Let $$S\subset\mathbb R^n$$ be a nonempty set. Given $$d\in [0,n)$$ and cube $$\overline{Q}\subset\mathbb with side length $$l=l(\overline{Q}) \in (0,1]$$ , we show that if the $$d$$ -Hausdorff content $$\mathcal H^d_{\infty}(\overline{Q}\cap S)$$ of set $$\overline{Q}\cap S$$ satisfies inequality S)<\overline{\lambda}l^{d}$$ for some $$\overline{\lambda}\in (0,1)$$ then $$\overline{Q}\setminus...

We show that a necessary and sufficient condition for the existence of a Kp,q factorization of the symmetric complete bipartite multi-digraph )"K:n n is (i) m = n == 0 (mod p) for p = q and (ii) m = n == a (mod d(p' + q')p'q'/e) for p =f. q, where d = (p,q), p' = p/d, q' = q/d, e = ()..,p'q').

Starting from an operator given as a product of q-exponential functions in irreducible representations the positive discrete series q-deformed algebra suq(1, 1), we express associated matrix elements terms d-orthogonal polynomials. An algebraic setting allows to establish some properties : recurrence relation, generating function, lowering operator, explicit expression and d-orthogonality relat...

Let $D\geq 1$ and $q\geq 3$ be two integers. $H(D)=H(D,q)$ denote the $D$-dimensional Hamming graph over a $q$-element set. ${\mathcal T}(D)$ Terwilliger algebra of $H(D)$. $V(D)$ standard T}(D)$-module. $\omega$ complex scalar. We consider unital associative $\mathfrak K_\omega$ defined by generators relations. The are $A$ $B$. relations $A^2 B-2 ABA +B A^2 =B+\omega A$, $B^2A-2 BAB+AB^2=A+\om...

It is proved that a code L(q) which monomially equivalent to the Pless symmetry C(q) of length $$2q+2$$ contains (0,1)-incidence matrix Hadamard 3- $$(2q+2,q+1,(q-1)/2)$$ design D(q) associated with Paley–Hadamard type II. Similarly, any ternary extended quadratic residue incidence 3-design I. If $$q=5, 11, 17, 23$$ , then full permutation automorphism group coincides D(q), and similar result h...