Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$, a vertex labeling $f : V(G)rightarrow mathbb{Z}_2$ induces an edge labeling $ f^{+} : E(G)rightarrow mathbb{Z}_2$ defined by $f^{+}(xy) = f(x) + f(y)$, for each edge $ xyin E(G)$.  For each $i in mathbb{Z}_2$, let $ v_{f}(i)=|{u in V(G) : f(u) = i}|$ and $e_{f^+}(i)=|{xyin E(G) : f^{+}(xy) = i}|$. A vertex labeling $f$ of a graph $G...

Equiangular tight frames (ETFs) have found significant applications in signal processing and coding theory due to their robustness to noise and transmission losses. ETFs are characterized by the fact that the coherence between any two distinct vectors is equal to the Welch bound. This guarantees that the maximum coherence between pairs of vectors is minimized. Despite their usefulness and wides...

Let $p$ be an analytic function defined on the open unit disc $mathbb{D}$ with $p(0)=1.$ The conditions on $alpha$ and $beta$ are derived for $p(z)$ to be subordinate to $1+4z/3+2z^{2}/3=:varphi_{C}(z)$ when $(1-alpha)p(z)+alpha p^{2}(z)+beta zp'(z)/p(z)$ is subordinate to $e^{z}$. Similar problems were investigated for $p(z)$ to lie in a region bounded by lemniscate of Bernoulli $|w^{2}-1|=1$ ...

A set is called a unique range set for a certain class of functions if each inverse image of that set uniquely determines a function from the given class. We show that a finite set is a unique range set, counting multiplicity, for non-Archimedean entire functions if and only if there is no non-trivial affine transformation preserving the set. Our proof uses a theorem of Berkovich to extend, to ...

If $f : S' \to S$ is a finite locally free morphism of schemes, we construct symmetric monoidal "norm" functor $f_\otimes \mathcal{H}_{\bullet}(S')\to \mathcal{H}_{\bullet}(S)$, where $\mathcal{H}_\bullet(S)$ the pointed unstable motivic homotopy category over $S$. $f$ étale, show that it stabilizes to \mathcal{S}\mathcal{H}(S') \mathcal{S}\mathcal{H}(S)$, $\mathcal{S}\mathcal{H}(S)$ $\mathbb{P...

Abstract We establish a version “over the ring” of celebrated Hilbert Irreducibility Theorem. Given finitely many polynomials in k + n {k+n} variables, with coefficients ? {\mathbb{Z}} , positive degree last n we show that if they are irreducible over and satisf...

We show that the algebraic K -theory of generalized archimedean valuation rings occurring in Durov’s compactification of the spectrum of a number ring is given by stable homotopy groups of certain classifying spaces.We also show that the “residue field at infinity” is badly behaved from a K -theoretic point of view.