Showing papers on "Coprime integers published in 2013"

Weight distributions of cyclic codes with respect to pairwise coprime order elements

Abstract: Let $\Bbb F_r$ be an extension of a finite field $\Bbb F_q$ with $r=q^m$. Let each $g_i$ be of order $n_i$ in $\Bbb F_r^*$ and $\gcd(n_i, n_j)=1$ for $1\leq i eq j \leq u$. We define a cyclic code over $\Bbb F_q$ by $$\mathcal C_{(q, m, n_1,n_2, ..., n_u)}=\{c(a_1, a_2, ..., a_u) : a_1, a_2, ..., a_u \in \Bbb F_r\},$$ where $$c(a_1, a_2, ..., a_u)=({Tr}_{r/q}(\sum_{i=1}^ua_ig_i^0), ..., {Tr}_{r/q}(\sum_{i=1}^ua_ig_i^{n-1}))$$ and $n=n_1n_2... n_u$. In this paper, we present a method to compute the weights of $\mathcal C_{(q, m, n_1,n_2, ..., n_u)}$. Further, we determine the weight distributions of the cyclic codes $\mathcal C_{(q, m, n_1,n_2)}$ and $\mathcal C_{(q, m, n_1,n_2,1)}$.

Results and conjectures on simultaneous core partitions

Abstract: An n-core partition is an integer partition whose Young diagram contains no hook lengths equal to n. We consider partitions that are simultaneously a-core and b-core for two relatively prime integers a and b. These are related to abacus diagrams and the combinatorics of the affine symmetric group (type A). We observe that self-conjugate simultaneous core partitions correspond to the combinatorics of type C, and use abacus diagrams to unite the discussion of these two sets of objects. In particular, we prove that (2n)- and (2mn+1)-core partitions correspond naturally to dominant alcoves in the m-Shi arrangement of type C_n, generalizing a result of Fishel--Vazirani for type A. We also introduce a major statistic on simultaneous n- and (n+1)-core partitions and on self-conjugate simultaneous (2n)- and (2n+1)-core partitions that yield q-analogues of the Coxeter-Catalan numbers of type A and type C. We present related conjectures and open questions on the average size of a simultaneous core partition, q-analogs of generalized Catalan numbers, and generalizations to other Coxeter groups. We also discuss connections with the cyclic sieving phenomenon and q,t-Catalan numbers.

Conjugacy action, induced representations and the Steinberg square for simple groups of Lie type

Abstract: Let $G$ be a finite simple group of Lie type, and let $\pi_G$ be the permutation representation of $G$ associated with the action of $G$ on itself by conjugation We prove that every irreducible representation of $G$ is a constituent of $\pi_G$, unless $G=PSU_n(q)$ and $n$ is coprime to $2(q+1)$, where precisely one irreducible representation fails Let St be the Steinberg representation of $G$ We prove that a complex irreducible representation of $G$ is a constituent of the tensor square $St\otimes St$, with the same exceptions as in the previous statement

Explicit Chabauty over number fields

Abstract: Let C be a smooth projective absolutely irreducible curve of genus g≥2 over a number field K of degree d, and let J denote its Jacobian. Let r denote the Mordell–Weil rank of J(K). We give an explicit and practical Chabauty-style criterion for showing that a given subset K⊆C(K) is in fact equal to C(K). This criterion is likely to be successful if r≤d(g−1). We also show that the only solution to the equation x2+y3=z10 in coprime nonzero integers is (x,y,z)=(±3,−2,±1). This is achieved by reducing the problem to the determination of K-rational points on several genus-2 curves where K=ℚ or ℚ(23) and applying the method of this paper.

Rational Associahedra and Noncrossing Partitions

Abstract: Each positive rational number $x>0$ can be written uniquely as $x=a/(b-a)$ for coprime positive integers $0 0$ a simplicial complex $\mathsf{Ass}(x)=\mathsf{Ass}(a,b)$ called the rational associahedron . It is a pure simplicial complex of dimension $a-2$, and its maximal faces are counted by the rational Catalan number $$\mathsf{Cat}(x)=\mathsf{Cat}(a,b):=\frac{(a+b-1)!}{a!\,b!}.$$The cases $(a,b)=(n,n+1)$ and $(a,b)=(n,kn+1)$ recover the classical associahedron and its "Fuss-Catalan" generalization studied by Athanasiadis-Tzanaki and Fomin-Reading. We prove that $\mathsf{Ass}(a,b)$ is shellable and give nice product formulas for its $h$-vector (the rational Narayana numbers ) and $f$-vector (the rational Kirkman numbers ). We define $\mathsf{Ass}(a,b)$ via rational Dyck paths : lattice paths from $(0,0)$ to $(b,a)$ staying above the line $y = \frac{a}{b}x$. We also use rational Dyck paths to define a rational generalization of noncrossing perfect matchings of $[2n]$. In the case $(a,b) = (n, mn+1)$, our construction produces the noncrossing partitions of $[(m+1)n]$ in which each block has size $m+1$.

A dynamical point of view on the set of B-free integers

Abstract: We extend the study of the square-free flow, recently introduced by Sarnak, to the more general context of B-free integers, that is to say integers with no factor in a given family B of pairwise relatively prime integers, the sum of whose reciprocals is finite. Relying on dynamical arguments, we prove in particular that the distribution of patterns in the characteristic function of the B-free integers follows a shift-invariant probability measure, and gives rise to a measurable dynamical system isomorphic to a specific minimal rotation on a compact group. As a by-product, we get the abundance of twin B-free integers. Moreover, we show that the distribution of patterns in small intervals also conforms to the same measure. When elements of B are squares, we introduce a generalization of the Mobius function, and discuss a conjecture of Chowla in this broader context.

Prime polynomials in short intervals and in arithmetic progressions

Abstract: In this paper we establish function field versions of two classical conjectures on prime numbers. The first says that the number of primes in intervals (x,x+x^epsilon] is about x^epsilon/log x and the second says that the number of primes p

The $y$-genus of the moduli space of $\operatorname{PGL}_{n}$-Higgs bundles on a curve (for degree coprime to $n$)

Abstract: Building on our previous joint work with A. Schmitt, we explain a recursive algorithm to determine the cohomology of moduli spaces of Higgs bundles on any given curve (in the coprime situation). As an application of the method, we compute the y -genus of the space of PGL n -Higgs bundles for any rank n , confirming a conjecture of T. Hausel.

Composed products and factors of cyclotomic polynomials over finite fields

Abstract: Let q = p s be a power of a prime number p and let $${\mathbb {F}_{\rm q}}$$ be a finite field with q elements. This paper aims to demonstrate the utility and relation of composed products to other areas such as the factorization of cyclotomic polynomials, construction of irreducible polynomials, and linear recurrence sequences over $${\mathbb {F}_{\rm q}}$$ . In particular we obtain the explicit factorization of the cyclotomic polynomial $${\Phi_{2^nr}}$$ over $${\mathbb {F}_{\rm q}}$$ where both r ? 3 and q are odd, gcd(q, r) = 1, and $${n\in \mathbb{N}}$$ . Previously, only the special cases when r = 1, 3, 5, had been achieved. For this we make the assumption that the explicit factorization of $${\Phi_r}$$ over $${\mathbb {F}_{\rm q}}$$ is given to us as a known. Let $${n = p_1^{e_1}p_2^{e_2}\cdots p_s^{e_s}}$$ be the factorization of $${n \in \mathbb{N}}$$ into powers of distinct primes p i , 1 ? i ? s. In the case that the multiplicative orders of q modulo all these prime powers $${p_i^{e_i}}$$ are pairwise coprime, we show how to obtain the explicit factors of $${\Phi_{n}}$$ from the factors of each $${\Phi_{p_i^{e_i}}}$$ . We also demonstrate how to obtain the factorization of $${\Phi_{mn}}$$ from the factorization of $${\Phi_n}$$ when q is a primitive root modulo m and $${{\rm gcd}(m, n) = {\rm gcd}(\phi(m),{\rm ord}_n(q)) = 1.}$$ Here $${\phi}$$ is the Euler's totient function, and ord n (q) denotes the multiplicative order of q modulo n. Moreover, we present the construction of a new class of irreducible polynomials over $${\mathbb {F}_{\rm q}}$$ and generalize a result due to Varshamov (Soviet Math Dokl 29:334---336, 1984).

Factoring 51 and 85 with 8 qubits

Abstract: We construct simplified quantum circuits for Shor's order-finding algorithm for composites N given by products of the Fermat primes 3, 5, 17, 257, and 65537. Such composites, including the previously studied case of 15, as well as 51, 85, 771, 1285, 4369,... have the simplifying property that the order of a modulo N for every base a coprime to N is a power of 2, significantly reducing the usual phase estimation precision requirement. Prime factorization of 51 and 85 can be demonstrated with only 8 qubits and a modular exponentiation circuit consisting of no more than four CNOT gates.

Klein forms and the generalized superelliptic equation

Abstract: If F (x;y)2 Z[x;y] is an irreducible binary form of degree k 3, then a theorem of Darmon and Granville implies that the generalized superelliptic equation F (x;y) = z l has, given an integer l maxf2; 7 kg, at most nitely many solutions in coprime integers x;y and z. In this paper, for large classes of forms of degree k = 3; 4; 6 and 12 (including, heuristically, \most" cubic forms), we extend this to prove a like result, where the parameter l is now taken to be variable. In the case of irreducible cubic forms, this provides the rst examples where such a conclusion has been proven. The method of proof combines classical invariant theory, modular Galois representations, and properties of elliptic curves with isomorphic mod-n Galois representations.

On a problem of Arnold: The average multiplicative order of a given integer

Abstract: For coprime integers g and n, let l(g) (n) denote the multiplicative order of g modulo n. Motivated by a conjecture of Arnold, we study the average of l(g) (n) as n <= x ranges over integers coprime to g, and x tending to infinity. Assuming the generalized Riemann Hypothesis, we show that this average is essentially as large as the average of the Carmichael lambda function. We also determine the asymptotics of the average of l(g) (p) as p <= x ranges over primes.

On the removal lemma for linear systems over Abelian groups

Abstract: In this paper we present an extension of the removal lemma to integer linear systems over abelian groups. We prove that, if the k-determinantal of an integer (kxm) matrix A is coprime with the order n of a group G and the number of solutions of the system Ax=b with x"1@?X"1,...,x"m@?X"m is o(n^m^-^k), then we can eliminate o(n) elements in each set to remove all these solutions.

Factoring 51 and 85 with 8 qubits

Abstract: We construct simplified quantum circuits for Shor's order-finding algorithm for composites N given by products of the Fermat primes 3, 5, 17, 257 and 65537. Such composites, including the previously studied case of 15, as well as 51, 85, 771, 1285, 4369, … have the simplifying property that the order of a modulo N for every base a coprime to N is a power of 2, significantly reducing the usual phase estimation precision requirement. Prime factorization of 51 and 85 can be demonstrated with only 8 qubits and a modular exponentiation circuit consisting of no more than four CNOT gates.

Consecutive primes in tuples

Abstract: In a recent advance towards the Prime $k$-tuple Conjecture, Maynard and Tao have shown that if $k$ is sufficiently large in terms of $m$, then for an admissible $k$-tuple $\mathcal{H}(x) = \{gx + h_j\}_{j=1}^k$ of linear forms in $\mathbb{Z}[x]$, the set $\mathcal{H}(n) = \{gn + h_j\}_{j=1}^k$ contains at least $m$ primes for infinitely many $n \in \mathbb{N}$. In this note, we deduce that $\mathcal{H}(n) = \{gn + h_j\}_{j=1}^k$ contains at least $m$ consecutive primes for infinitely many $n \in \mathbb{N}$. We answer an old question of Erd\H os and Turan by producing strings of $m + 1$ consecutive primes whose successive gaps $\delta_1,\ldots,\delta_m$ form an increasing (resp. decreasing) sequence. We also show that such strings exist with $\delta_{j-1} \mid \delta_j$ for $2 \le j \le m$. For any coprime integers $a$ and $D$ we find arbitrarily long strings of consecutive primes with bounded gaps in the congruence class $a \bmod D$.

Loop-shaping Robust Control

Abstract: Introduction ix Chapter 1 The Loop-shaping Approach 1 1.1 Principle of the method 1 1.2 Generalized phase and gain margins 14 1.3 Limitations inherent to bandwidth 17 1.4 Examples 18 1.5 Conclusion 30 Chapter 2 Loop-shaping H Synthesis 33 2.1 The formalism of coprime factorizations 33 2.2 Robustness of normalized coprime factor plant descriptions 42 2.3 Explicit solution of the problem of robust stabilization of coprime factor plant descriptions 54 2.4 Robustness and -gap 77 2.5 Loop-shaping synthesis approach 82 2.6 Discrete approach 120 Chapter 3 Two Degrees-of-Freedom Controllers 135 31 Principle 135 3.2 Two-step approach 143 3.3 One-step approach 156 3.4 Comparison of the two approaches 165 3.5 Example 166 3.6 Compensation for a measurable disturbance at the model s output 174 Chapter 4 Extensions and Optimizations 187 4.1 Introduction 187 4.2 Fixed-order synthesis 188 4.4 Towards a new approach to loop-shaping fixed-order controller synthesis, etc 242 APPENDICES 245 Appendix 1 247 Appendix 2 251 Bibliography 255 Index 259

On the probability distribution of the gcd and lcm of $r$-tuples of integers

Abstract: This paper is devoted to the study of statistical properties of the greatest common divisor and the least common multiple of random samples of positive integers.

Strongly primitive species with potentials I: Mutations

Abstract: Motivated by the mutation theory of quivers with potentials developed by Derksen-Weyman-Zelevinsky, and the representation-theoretic approach to cluster algebras it provides, we propose a mutation theory of species with potentials for species that arise from skew-symmetrizable matrices that admit a skew-symmetrizer with pairwise coprime diagonal entries. The class of skew-symmetrizable matrices covered by the mutation theory proposed here contains a class of matrices that do not admit global unfoldings, that is, unfoldings compatible with all possible sequences of mutations.

The Probability That Random Positive Integers Are 3-Wise Relatively Prime

Abstract: A list of positive integers are 3-wise relatively prime if every three of them are relatively prime. In this note we consider the problem of finding the probability that k positive integers are 3-wise relatively prime and give an exact formula for this probablility.

Coprime subdegrees for primitive permutation groups and completely reducible linear groups

Abstract: In this paper we answer a question of Gabriel Navarro about orbit sizes of a finite linear group H ⊆ GL(V) acting completely reducibly on a vector space V: if the H-orbits containing the vectors a and b have coprime lengths m and n, we prove that the H-orbit containing a + b has length mn. Such groups H are always reducible if n,m > 1. In fact, if H is an irreducible linear group, we show that, for every pair of non-zero vectors, their orbit lengths have a non-trivial common factor.

3d superconformal indices and isomorphisms of M2-brane theories

Abstract: We test several expected isomorphisms between the U(N) × U(N) ABJM theory and (SU(N) × SU(N))/ $ {{\mathbb{Z}}_N} $ theory including the BLG theory by comparing their superconformal indices. From moduli space analysis, it is expected that this equivalence can hold if and only if the rank N and Chern-Simons level k are coprime. We also calculate the index of the ABJ theory and investigate whether some theories with identical moduli spaces are isomorphic or not.

Conciseness of coprime commutators in finite groups

Abstract: Let $G$ be a finite group. We show that the order of the subgroup generated by coprime $\gamma_k$-commutators (respectively $\delta_k$-commutators) is bounded in terms of the size of the set of coprime $\gamma_k$-commutators (respectively $\delta_k$-commutators). This is in parallel with the classical theorem due to Turner-Smith that the words $\gamma_k$ and $\delta_k$ are concise.

Computing upper cluster algebras

Abstract: This paper develops techniques for producing presentations of upper cluster algebras. These techniques are suited to computer implementation, and will always succeed when the upper cluster algebra is totally coprime and finitely generated. We include several examples of presentations produced by these methods.

Coprime solutions to ax≡b (mod n)

Abstract: Abstract. It is well known that a congruence ax≡b (mod n) has a solution if and only if , and, if the condition is satisfied, the number of incongruent solutions equals . In 2010, Alomair, Clark and Poovendran proved that the congruence ax≡b (mod n) has a solution coprime to n if and only if , as an auxiliary result playing a key role in a problem related to an electronic signature. In this paper we provide a concise proof of this result, together with a closed formula for the number of incongruent solutions coprime to n as well. Moreover, a bound is presented for the probability that, for randomly chosen , this congruence possesses at least one solution coprime to n.

Prime labeling for some helm related graphs

Abstract: A graph with vertex set V is said to have a prime labeling if its vertices are labeled with distinct integers 1,2,3 … 𝑉 such that for edge 𝑥𝑦 the labels assigned to x and y are relatively prime. A graph which admits prime labeling is called a prime graph. In this paper we investigate prime labeling for some helm related graphs. We also discuss prime labeling in the context of some graph operations namely fusion and duplication in Helm 𝐻𝑛