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...

Abstract Let $P_1,P_2,\dots , P_k$ be complex polynomials of degree at least two that are not simultaneously conjugate to monomials or Chebyshev polynomials, and $S$ the semigroup under composition generated by P_k$. We show all elements share a measure maximal entropy if only intersection principal left ideals $SP_1\cap SP_2\cap \dots \cap SP_k$ is non-empty.

We consider the classical momentum- or velocity-dependent two-dimensional Hamiltonian given by $$\mathcal H_N = p_1^2 + p_2^2 +\sum_{n=1}^N \gamma_n(q_1 p_1 q_2 p_2)^n ,$$ where $q_i$ and $p_i$ are generic canonical variables, $\gamma_n$ arbitrary coefficients, $N\in \mathbb N$. For $N=2$, being both $\gamma_1,\gamma_2$ different from zero, this reduces to Zernike system. prove that $\mathcal H...

Let $[\, \cdot\,]$ be the floor function. In this paper, we prove that when $1<c<\frac{16559}{15276}$, then every sufficiently large positive integer $N$ can represented in form \begin{equation*} N=[p^c_1]+[p^c_2]+[p^c_3]\,, \end{equation*} where $p_1,p_2,p_3$ are primes, such $p_1=x^2 + y^2 +1$.

Phylogenetic trees canonically arise as embeddings of phylogenetic networks. We recently showed that the problem deciding if two networks embed same sets is computationally hard, \blue{in particular, we it to be $\Pi^P_2$-complete}. In this paper, establish a polynomial-time algorithm for decision initial consists normal network and tree-child network. The running time quadratic in size leaf se...

Harmonic centrality calculates the importance of a node in network by adding inverse geodesic distances this to all other nodes. centralization, on hand, is graph-level score based node-level harmonic centrality. In paper, we present some results both and centralization graphs resulting from graph products such as Cartesian direct path $P_2$ with any $P_m$, cycle $C_m$, fan $F_m$ graphs.