Like ants, termites are entirely eusocial and have profound ecological significance in the tropics. Following upon recent studies reporting more than a quarter of all known fossil termites, we present first phylogeny termite lineages using exemplar Cretaceous, Tertiary, Recent taxa. Relationships among families were largely unaffected by addition extinct taxa, but analysis revealed extensive grades stem-group taxa divergence some modern Cretaceous. Rhinotermitidae, Serritermitidae, "higher"...

A comprehensive compendium on the taxonomy and biology of 3106 living fossil species worlds termites is presented, along with reviews Isoptera morphology evolution, identification keys, history termite systematics, summary 363 significant pest species. complete bibliography provided nearly 5000 references covering virtually all aspects through December 2011.The thoroughly reviewed illustrated original scanning electron micrographs photomicrographs, cuticular anatomy those internal organs...

ABSTRACT A comprehensive compendium on the taxonomy and biology of 3106 living fossil species worlds termites is presented, along with reviews Isoptera morphology evolution, identification keys, history termite systematics, summary 363 significant pest species. complete bibliography provided nearly 5000 references covering virtually all aspects through December 2011. The thoroughly reviewed illustrated original scanning electron micrographs photomicrographs, cuticular anatomy those internal...

In 1974, Welch derived lower bounds (known as bounds) on the maximum of modulus inner products distinct elements in a finite collection unit vectors dimensional Hilbert space.Recently, continuous are for Bessel family indexed over measure spaces space.In this paper, we derive both discrete and Banach which contain space case particular case.We formulate several problems future research.

Thirty-nine available family-group names are identified within the insect order Isoptera (termites). For all correct author, date, type genus, and combining stem provided for first time. This nomenclatural compilation is done to stabilize usage of family- group in advance a world catalog. Several problems priority discussed. The little understood subfamily Foraminitermitinae diagnosed; while generally believed by many authors be new, unnamed subfamily, it was fact established Holmgren nearly...

The termite species from Brazil's Early Cretaceous (Aptian-aged) Crato (Santana) Formation are evaluated on the basis of degree character variation seen in modern species, using a series 56 specimens, scanning electron microscopy minute structures, and bivariate plot proportional sizes sclerotized body structures. Of previously described only following considered valid: Mariconitermes talicei Fontes Vulcano, Meiatermes araripena Krishna, Cratomastotermes wolfschwenningeri Bechly,...

The most diverse and best-preserved paleofauna of the higher termites heretofore known, all found in Miocene amber Dominican Republic, is described. imago Coptotermes priscus Emerson redescribed, soldier C. priscus, first known fossil this genus, fauna includes following 29 new species, existing genera, with Krishna Grimaldi as authors each: Rhinotermitidae, two species based on imagoes each—Coptotermes hirsutus paleodominicanus; Termitidae, 23 imagoes—Amitermes lucidus, Anoplotermes bohio,...

Let $\mathcal{E}$ be a Hilbert C*-module over unital C*-algebra $\mathcal{A}$. $A: \mathcal{D}(A) \subseteq \mathcal{E} \to \mathcal{E}$ and $B: \mathcal{D}(B)\subseteq \mathcal{E}\to possibly unbounded self-adjoint morphisms. Then for all $x \in \mathcal{D}(AB)\cap \mathcal{D}(BA)$ with $\langle x, x \rangle =1$, we show that \begin{align*} (1) \quad \Delta _x(B)^2d_x(A)^2+\Delta _x(A)^2d_x(B)^2\geq \frac{(\langle \{A,B\}x, -\{\langle Ax, \rangle,\langle Bx, \rangle\})^2-(\langle [A,B]x,...

Let $\mathcal{E}$ be a Hilbert C*-module over unital C*-algebra $\mathcal{A}$. $A: \mathcal{D}(A) \subseteq \mathcal{E} \to \mathcal{E}$ and $B: \mathcal{D}(B)\subseteq \mathcal{E}\to possibly unbounded self-adjoint morphisms. Then for all $x \in \mathcal{D}(AB)\cap \mathcal{D}(BA)$ with $\langle x, x \rangle =1$, we show that \begin{align*} (1) \quad\quad\quad d_x(A)^2+d_x(B)^2&\geq \mp \langle \{A,B\}x, +\langle (A\pm B)x, y \rangle\langle y, B)x \rangle\pm \{\langle Ax, \rangle,...

Let $\mathcal{E}$ be a Hilbert C*-module over unital C*-algebra $\mathcal{A}$. $A: \mathcal{D}(A) \subseteq \mathcal{E} \to \mathcal{E}$ and $B: \mathcal{D}(B)\subseteq \mathcal{E}\to possibly unbounded self-adjoint morphisms. Then for all $x \in \mathcal{D}(AB)\cap \mathcal{D}(BA)$ with $\langle x, x \rangle =1$, we show that \begin{align*} (1) \quad \Delta _x(B)^2d_x(A)^2+\Delta _x(A)^2d_x(B)^2\geq \frac{(\langle \{A,B\}x, -\{\langle Ax, \rangle,\langle Bx, \rangle\})^2-(\langle [A,B]x,...

Let $\mathcal{X}$ be a p-adic Hilbert space. $A: \mathcal{D}(A)\subseteq \mathcal{X}\to \mathcal{X}$ and $B: \mathcal{D}(B)\subseteq possibly unbounded linear operators. For $x \in \mathcal{D}(A)$ with $\langle x, x \rangle =1$, define $ \Delta _x(A):= \|Ax- \langle Ax, \|.$ Then for all \mathcal{D}(AB)\cap \mathcal{D}(BA)$ we show that \begin{align*} (1) \quad \max\{\Delta_x(A), \Delta_x(B)\}\geq \frac{\sqrt{\bigg|\big\langle [A,B]x, \big\rangle ^2+\big(\langle \{A,B\}x, -2\langle...

We introduce the notion of noncommutative equiangular lines and derive versions fundamental van Lint-Seidel relative Gerzon universal bounds.

Dilation of

We prove the following two results. \begin{enumerate} \item Let $\mathcal{A}$ be a unital commutative C*-algebra and $\mathcal{A}^d$ standard Hilbert C*-module over $\mathcal{A}$. $n\geq d$. If $\{\tau_j\}_{j=1}^n$ is any collection of vectors in such that $\langle \tau_j, \tau_j \rangle =1$, $\forall 1\leq j \leq n$, then \begin{align*} \max _{1\leq j,k n, j\neq k}\|\langle \tau_k\rangle ||^{2m}\geq \frac{1}{n-1}\left[\frac{n}{{d+m-1\choose m}}-1\right], \quad \forall m \in \mathbb{N}....

Let (Ω, µ), (∆, ν) be measure spaces. ({fα}α∈Ω, {τα}α∈Ω) and ({gβ}β∈∆, {ωβ}β∈∆) continuous p-Schauder frames for a Banach space X . Then every x ∈ \ {0}, we show that(1)µ(supp(θf x))1⁄pν(supp(θgx))1⁄q ≥ 1⁄sup α∈Ω,β∈∆ |fα(ωβ )|, ν(supp(θgx)) µ(supp(θf x))1⁄q |gβ (τα)|whereθf : ::l 1→ θf Lᵖ(Ω, µ); Ω α (θf x)(α) := fα(x) K,θg θgx Lᵖ(∆, ν); ∆ β (θgx)(β) gβ(x) Kand q is the conjugate index of p. We call Inequality (1) as Functional Continuous Uncertainty Principle. It improves...

A new subfamily, genus, and species, Archeorhinotermitinae, Archeorhinotermes rossi, from Burmese amber, dated as Turonian-Cenomanian (90–100 mya) of the Cretaceous period, are described figured. Comparisons made between other subfamilies Rhinotermitidae subfamily. This is first fossil record family Cretaceous.