Let $p\geq 2$ be a large prime, and let $N\gg ( \log p)^{1+\varepsilon}$. This note proves the existence of primitive roots in the short interval $[M,M+N]$, where $M \geq 2$ is a fixed number, and $ \varepsilon>0$ is a small number. In particular, th...
This note presents an upper bound for the least prime primitive roots $g^*(p)$ modulo $p$, a large prime. The current literature has several estimates of the least prime primitive root $g^*(p)$ modulo a prime $p\geq 2$ such as $g^*(p)\ll p^c, c>2.8$....
Let $p \geq 2$ be a large prime, and let $k \ll \log p $ be a small integer. This note proves the existence of various configurations of $(k+1)$-tuples of consecutive and quasi consecutive primitive roots $n+a_0, n+a_1, n+a_2, \ldots, n+a_k$ in the f...
We examine linear sums of primitive roots and their inverses in finite fields. In particular, we refine a result by Li and Han, and show that every $p> 13$ has a pair of primitive roots $a$ and $b$ such that $a+ b$ and $a^{-1} + b^{-1}$ are also prim...
The Chinese Roots of Linear Algebra by Roger Hart chronicles the linear problems of ancient China in the Nine Chapters of the Mathematical Art, and supplies new insights about their solution....
I believe that I am projected because there’s a bit of an overhang. I did cup my band so you can see where my roots are. I sometimes feel like I have tall or wide roots which makes me think I have s...
The $n^{th}$ cyclotomic polynomial $Φ_n(x)$ is the minimal polynomial of an $n^{th}$ primitive root of unity. Hence $Φ_n(x)$ is trivially zero at primitive $n^{th}$ roots of unity. Using finite Fourier analysis we derive a formula for $Φ_n(x)$ at...
We give deterministic polynomial-time algorithms that, given an order, compute the primitive idempotents and determine a set of generators for the group of roots of unity in the order. Also, we show that the discrete logarithm problem in the group of...
blowers are commonly referred to as air blowers or PD (positive displacement) blowers. The Roots-type blower is simple and widely used. It can be more
In this paper, we research more in depth properties of Backtracking New Q-Newton's method (recently designed by the third author), when used to find roots of meromorphic functions. If $f=P/Q$, where $P$ and $Q$ are polynomials in 1 complex variable...
Psammophilous plants are special plants that flourish in sand moving environments. There are two main mechanisms by which the wind affects these plants: (i) sand drift exposes roots and covers branches--the exposed roots turn into new plants and the...
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A time-dependent monic polynomial in the z variable with N distinct roots such that exactly one root has multiplicity m>=2 is considered. For k=1,2, the k-th derivatives of the N roots are expressed in terms of the derivatives of order j<= k of the f...
Hello. My language has a tri-vocalic root system and yesterday I went through my vowels *again* and took it down to 10 from 14, reducing my stock of roots from 2744 to 1000. If I disallow the same vow...
This is an exposition of some basic mathematical aspects of quantum logic gates. At first we established some general formulas for the case of arbitrary quantum gate A with unique restriction A^2=I. The explicit form of the generators and roots of...
released an eponymous album in 1975, The Grass Roots, containing the single "Mamacita", which charted at number 71 on the Billboard Hot 100. Follow-up singles
We use the theory of logarithmic line bundles to construct compactifications of spaces of roots of a line bundle on a family of curves, generalising work of a number of authors. This runs via a study of the torsion in the tropical and logarithmic jac...
After reconsidering the theorem of continuity of the roots of a polynomial in terms of its coefficients in the deformation framework, we study the stability of the greater common divisor of two polynomials compared to perturbations on their roots. We...
For each of the exceptional Lie superalgebras with indecomposable Cartan matrix, we give the explicit list of its roots of and the corresponding Chevalley basis for one of the inequivalent Cartan matrices, the one corresponding to the greatest number...
What began as a vacation to a town in Mexico became a round trip journey between two countries, marked by love, a painful loss, and much waiting. For years, Yesica Aramburo and Ramón Vega sustained their relationship between airports, immigration processes, and grief, until an ap…