scholarly journals On Non-Archimedean Curves Omitting Few Components and their Arithmetic Analogues

2017 ◽  
Vol 69 (1) ◽  
pp. 130-142 ◽  
Author(s):  
Aaron Levin ◽  
Julie Tzu-Yueh Wang
Keyword(s):  
Projective Variety ◽  
Quadratic Field ◽  
Closed Field ◽  
Necessary And Sufficient Condition ◽  
Integral Points ◽  
Absolute Value ◽  
Geometric Conditions ◽  
Necessary And Sufficient ◽  
Analytic Maps ◽  
Nef Divisors

AbstractLet k be an algebraically closed field completewith respect to a non-Archimedean absolute value of arbitrary characteristic. Let D1 , … , Dn be effective nef divisors intersecting transversally in an n-dimensional nonsingular projective variety X. We study the degeneracy of non-Archimedean analytic maps from k into under various geometric conditions. When X is a rational ruled surface and D1 and D2 are ample, we obtain a necessary and sufficient condition such that there is no non-Archimedean analytic map from k into . Using the dictionary between non-Archimedean Nevanlinna theory and Diophantine approximation that originated in earlier work with T. T. H. An, we also study arithmetic analogues of these problems, establishing results on integral points on these varieties over ℤ or the ring of integers of an imaginary quadratic field.

Prime Producing Quadratic Polynomials and Class-Numbers of Real Quadratic Fields

1990 ◽  
Vol 42 (2) ◽  
pp. 315-341 ◽  
Author(s):  
Stéphane Louboutin
Keyword(s):  
Class Number ◽  
Quadratic Field ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Quadratic Polynomials ◽  
Real Quadratic Fields ◽  
Consecutive Integers ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Real Quadratic

Frobenius-Rabinowitsch's theorem provides us with a necessary and sufficient condition for the class-number of a complex quadratic field with negative discriminant D to be one in terms of the primality of the values taken by the quadratic polynomial with discriminant Don consecutive integers (See [1], [7]). M. D. Hendy extended Frobenius-Rabinowitsch's result to a necessary and sufficient condition for the class-number of a complex quadratic field with discriminant D to be two in terms of the primality of the values taken by the quadratic polynomials and with discriminant D (see [2], [7]).

scholarly journals Cuspidal and noncuspidal robot manipulators

Robotica ◽  
2007 ◽  
Vol 25 (6) ◽  
pp. 677-689 ◽  
Author(s):  
Philippe Wenger
Keyword(s):  
Robot Manipulators ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Geometric Conditions ◽  
Necessary And Sufficient ◽  
Full Classification

SUMMARYThis article synthezises the most important results on the kinematics of cuspidal manipulators i.e. nonredundant manipulators that can change posture without meeting a singularity. The characteristic surfaces, the uniqueness domains and the regions of feasible paths in the workspace are defined. Then, several sufficient geometric conditions for a manipulator to be noncuspidal are enumerated and a general necessary and sufficient condition for a manipulator to be cuspidal is provided. An explicit DH-parameter-based condition for an orthogonal manipulator to be cuspidal is derived. The full classification of 3R orthogonal manipulators is provided and all types of cuspidal and noncuspidal orthogonal manipulators are enumerated. Finally, some facts about cuspidal and noncuspidal 6R manipulators are reported.

Approximations of small jumps of Lévy processes with a view towards simulation

2001 ◽  
Vol 38 (2) ◽  
pp. 482-493 ◽  
Author(s):  
Søren Asmussen ◽  
Jan Rosiński
Keyword(s):  
Levy Processes ◽  
Error Rates ◽  
Levy Process ◽  
Series Expansions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Compound Poisson ◽  
Absolute Value ◽  
In Series ◽  
Necessary And Sufficient

Let X = (X(t):t ≥ 0) be a Lévy process and X∊ the compensated sum of jumps not exceeding ∊ in absolute value, σ2(∊) = var(X∊(1)). In simulation, X - X∊ is easily generated as the sum of a Brownian term and a compound Poisson one, and we investigate here when X∊/σ(∊) can be approximated by another Brownian term. A necessary and sufficient condition in terms of σ(∊) is given, and it is shown that when the condition fails, the behaviour of X∊/σ(∊) can be quite intricate. This condition is also related to the decay of terms in series expansions. We further discuss error rates in terms of Berry-Esseen bounds and Edgeworth approximations.

scholarly journals Nearly Gorenstein cyclic quotient singularities

2020 ◽  
Author(s):  
Alessio Caminata ◽  
Francesco Strazzanti
Keyword(s):  
Cyclic Group ◽  
Closed Field ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Algebraically Closed Field ◽  
Quotient Singularities ◽  
Necessary And Sufficient ◽  
Cyclic Quotient Singularities

Abstract We investigate the nearly Gorenstein property among d-dimensional cyclic quotient singularities $$\Bbbk \llbracket x_1,\dots ,x_d\rrbracket ^G$$ k 〚 x 1 , ⋯ , x d 〛 G , where $$\Bbbk $$ k is an algebraically closed field and $$G\subseteq {\text {GL}}(d,\Bbbk )$$ G ⊆ GL ( d , k ) is a finite small cyclic group whose order is invertible in $$\Bbbk $$ k . We prove a necessary and sufficient condition to be nearly Gorenstein that also allows us to find several new classes of such rings.

scholarly journals Spectral Properties of Unitary Cayley Graphs of Finite Commutative Rings

10.37236/2390 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Xiaogang Liu ◽  
Sanming Zhou
Keyword(s):  
Spectral Properties ◽  
Line Graph ◽  
Commutative Rings ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Absolute Value ◽  
Vertex Set ◽  
Finite Commutative Ring ◽  
Unitary Cayley Graph ◽  
Necessary And Sufficient

Let $R$ be a finite commutative ring. The unitary Cayley graph of $R$, denoted $G_R$, is the graph with vertex set $R$ and edge set $\left\{\{a,b\}:a,b\in R, a-b\in R^\times\right\}$, where $R^\times$ is the set of units of $R$. An $r$-regular graph is Ramanujan if the absolute value of every eigenvalue of it other than $\pm r$ is at most $2\sqrt{r-1}$. In this paper we give a necessary and sufficient condition for $G_R$ to be Ramanujan, and a necessary and sufficient condition for the complement of $G_R$ to be Ramanujan. We also determine the energy of the line graph of $G_R$, and compute the spectral moments of $G_R$ and its line graph.

scholarly journals On the Irreducibility of Fourth Dimensional Tuba's Representation of the Pure Braid Group on Three Strands

2018 ◽  
Vol 11 (3) ◽  
pp. 682-701
Author(s):  
Hasan A. Haidar ◽  
Mohammad N. Abdulrahim
Keyword(s):  
Braid Group ◽  
Sufficient Conditions ◽  
Complex Numbers ◽  
Closed Field ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Algebraically Closed Field ◽  
Pure Braid Group ◽  
Necessary And Sufficient

We consider Tuba's representation of the pure braid group, $%P_{3} $, given by the map $\phi :P_{3}\longrightarrow GL(4,F)$, where $F$ is an algebraically closed field. After, specializing the indeterminates used in defining the representation to non- zero complex numbers, we find sufficient conditions that guarantee the irreducibility of Tuba's representation of the pure braid group $P_{3}$ with dimension $d=4$. Under further restriction for the complex specialization of the indeterminates, we get a necessary and sufficient condition for the irreducibility of $\phi

Approximations of small jumps of Lévy processes with a view towards simulation

2001 ◽  
Vol 38 (02) ◽  
pp. 482-493 ◽  
Author(s):  
Søren Asmussen ◽  
Jan Rosiński
Keyword(s):  
Levy Processes ◽  
Error Rates ◽  
Levy Process ◽  
Series Expansions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Compound Poisson ◽  
Absolute Value ◽  
In Series ◽  
Necessary And Sufficient

Let X = (X(t):t ≥ 0) be a Lévy process and X ∊ the compensated sum of jumps not exceeding ∊ in absolute value, σ2(∊) = var(X ∊(1)). In simulation, X - X ∊ is easily generated as the sum of a Brownian term and a compound Poisson one, and we investigate here when X ∊/σ(∊) can be approximated by another Brownian term. A necessary and sufficient condition in terms of σ(∊) is given, and it is shown that when the condition fails, the behaviour of X ∊/σ(∊) can be quite intricate. This condition is also related to the decay of terms in series expansions. We further discuss error rates in terms of Berry-Esseen bounds and Edgeworth approximations.

scholarly journals Use of the rabinowitsch polynomial to determine the class groups of a real quadratic field

1994 ◽  
Vol 50 (3) ◽  
pp. 435-443
Author(s):  
R.A. Mollin
Keyword(s):  
Class Group ◽  
Quadratic Field ◽  
Sufficient Condition ◽  
Real Quadratic Field ◽  
Necessary And Sufficient Condition ◽  
Class Groups ◽  
Necessary And Sufficient ◽  
Real Quadratic

The main result is a necessary and sufficient condition for the class group of a real quadratic field to be determined by primality properties of the well-known Rabinowitsch polynomial.

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