we characterize strictly diagonal type of embeddings offinitary symmetric groups in terms of cardinality and the characteristic. namely, we prove thefollowing.let $kappa$ be an infinite cardinal. if$g=underset{i=1}{stackrel{infty}bigcup} g_i,$ where $ g_i=fsym(kappa n_i)$,$(h=underset{i=1}{stackrel{infty}bigcup}h_i, $ where $ h_i=alt(kappa n_i) ), $ is a group of strictly diagonal type and$xi=(...

In CRYPTO’03, Patarin conjectured a lower bound on the number of distinct solutions $$(P_1, \ldots , P_{q}) \in (\{0,1\}^{n})^{q}$$ satisfying system equations form $$X_i \oplus X_j = \lambda _{i,j}$$ such that $$P_1, P_2, $$ $$P_{q}$$ are pairwise distinct. This result is known as “ $$P_i P_j$$ Theorem for any $$\xi _{\max }$$ ” or alternatively Mirror Theory general which was later proved by ...

We study the class of all finite directed graphs up to primitive positive constructability. The resulting order has a unique greatest element, namely graph $P_1$ with one vertex and no edges. lower bound, $P_2$ two vertices edge. Our main result is complete description bounds $P_2$; we call these submaximal. show that every not equivalent below submaximal graphs.

Graph theory is a branch of algebra that growing rapidly both in concept and application studies. This graph can be used chemistry, transportation, cryptographic problems, coding theory, design communication network, etc. There currently bridge between graphs algebra, especially an algebraic structures, namely algebra. One researchs on formed by prime ring elements or called over R. The commuta...

We study the boundedness of commutators bi-parameter singular integrals between mixed spaces $$ [b,T]: L^{p_1}L^{p_2} \to L^{q_1}L^{q_2} in off-diagonal situation $q_i,p_i\in(1,\infty)$ where we also allow $q_i\not= p_i.$ Boundedness is fully characterized for several arrangements integrability exponents with some open problems presented.

Abstract In this paper, we study limiting embeddings of Besov-type and Triebel-Lizorkin-type spaces, $$\text {id}_\tau : {B}_{p_1,q_1}^{s_1,\tau _1}(\Omega ) \hookrightarrow {B}_{p_2,q_2}^{s_2,\tau _2}(\Omega )$$ id τ : B</mml...

Abstract A $k$-query locally decodable code (LDC) $C$ allows one to encode any $n$-symbol message $x$ as a codeword $C(x)$ of $N$ symbols such that each symbol can be recovered by looking at $k$ $C(x)$, even if constant fraction has been corrupted. Currently, the best known LDCs are matching vector codes (MVCs). modulus $m=p_1^{\alpha _1}p_2^{\alpha _2}\cdots p_r^{\alpha _r}$ may result in an M...