Showing papers on "Coprime integers published in 2014"

The Exponential Diophantine Equation2x+by=cz

Abstract: Let and be fixed coprime odd positive integers with . In this paper, a classification of all positive integer solutions of the equation is given. Further, by an elementary approach, we prove that if , then the equation has only the positive integer solution , except for and , where is a positive integer with .

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 index statistic on simultaneous n- and (n+1)-core partitions and on self-conjugate simultaneous 2n- and (2n+1)-core partitions that yield q-analogs 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.

Doa estimation exploiting coprime arrays with sparse sensor spacing

Abstract: In this paper, we propose effective coprime array configurations in which the minimum interelement spacing is much larger than the typical half-wavelength requirement. Such configurations are important in many applications where the half-wavelength requirement cannot be met due to the physical sensors size or to avoid spatial oversampling in wideband operations. The application of such coprime arrays in direction-of-arrival estimations is examined using different algorithms.

Rational parking functions and Catalan numbers

Abstract: The classical parking functions, counted by the Cayley number (n+1)^(n-1), carry a natural permutation representation of the symmetric group S_n in which the number of orbits is the n'th Catalan number. In this paper, we will generalize this setup to rational parking functions indexed by a pair (a,b) of coprime positive integers. We show that these parking functions, which are counted by b^(a-1), carry a permutation representation of S_a in which the number of orbits is a rational Catalan number. We compute the Frobenius characteristic of the S_a-module of (a,b)-parking functions. Next we propose a combinatorial formula for a q-analogue of the rational Catalan numbers and relate this formula to a new combinatorial model for q-binomial coefficients. Finally, we discuss q,t-analogues of rational Catalan numbers and parking functions (generalizing the shuffle conjecture for the classical case) and present several conjectures.

Model Reduction by Rational Interpolation

Abstract: The last two decades have seen major developments in inter- polatory methods for model reduction of large-scale linear dynamical systems. Advances of note include the ability to produce (locally) opti- mal reduced models at modest cost; rened methods for deriving inter- polatory reduced models directly from input/output measurements; and extensions for the reduction of parametrized systems. This chapter oers a survey of interpolatory model reduction methods starting from basic principles and ranging up through recent developments that include weighted model reduction and structure-preserving methods based on generalized coprime representations. Our discussion is supported by an assortment of numerical examples. Mathematics Subject Classication (2010). 41A05, 93A15, 93C05, 37M99. Keywords. Rational interpolation, model reduction,H2 approximation, parametric systems, generalized coprime factorization, weighted model reduction, Loewner framework

Compositional (km,kn)-Shuffle Conjectures

Abstract: In 2008, Haglund, Morse and Zabrocki formulated a Compositional form of the Shuffle Conjecture of Haglund et al. In very recent work, Gorsky and Negut by combining their discoveries with the work of Schiffmann-Vasserot on the symmetric function side and the work of Hikita and Gorsky-Mazin on the combinatorial side, were led to formulate an infinite family of conjectures that extend the original Shuffle Conjecture of Haglund et al. In fact, they formulated one conjecture for each pair (m,n) of coprime integers. This work of Gorsky-Negut leads naturally to the question as to where the Compositional Shuffle Conjecture of Haglund-Morse-Zabrocki fits into these recent developments. Our discovery here is that there is a compositional extension of the Gorsky-Negut Shuffle Conjecture for each pair (km,kn), with (m,n) co-prime and k > 1.

Shifts of generators and delta sets of numerical monoids

Abstract: Let S be a numerical monoid with minimal generating set 〈n1, …, nt〉. For m ∈ S, if $m = \sum_{i = 1}^{t} x_{i}n_{i}$, then $\sum_{i = 1}^{t} x_{i}$ is called a factorization length of m. We denote by ℒ(m) = {m1, …, mk} (where mi N then |Δ(Mn)| = 1. If t = 2 and r1 and r2 are relatively prime, then we determine a value for N which is sharp.

DOA estimation exploiting coprime frequencies

Abstract: Coprime array, which utilizes a coprime pair of uniform linear subarrays, is an attractive structure to achieve sparse array configurations. Alternatively, effective coprime array configurations can be implemented using a uniform linear array with two coprime sensing frequencies. This enables the integration of the coprime array and filter concepts to achieve high capabilities in meeting system performance and complexity constraints. This paper examines its performance for direction-of-arrival estimations. In particular, we analyze the number of detectable signals and the estimation accuracy as related to the array configurations and sensing frequencies.

Fusion hierarchies, T-systems, and Y-systems of logarithmic minimal models

Abstract: A Temperley?Lieb (TL) loop model is a Yang?Baxter integrable lattice model with nonlocal degrees of freedom. On a strip of width , the evolution operator is the double-row transfer tangle D(u), an element of the TL algebra TLN(?) with loop fugacity ? = 2cos?, . Similarly, on a cylinder, the single-row transfer tangle T(u) is an element of the so-called enlarged periodic TL algebra. The logarithmic minimal models comprise a subfamily of the TL loop models for which the crossing parameter ? = (p?? p)?/p? is a rational multiple of ? parameterized by coprime integers 1 ? p 2 takes the form of functional relations for D(u) and T(u) of polynomial degree p?. These derive from fusion hierarchies of commuting transfer tangles Dm, n(u) and Tm, n(u), where D(u) = D1,1(u) and T(u) = T1,1(u). The fused transfer tangles are constructed from (m, n)-fused face operators involving Wenzl?Jones projectors Pk on k = m or k = n nodes. Some projectors Pk are singular for k ? p?, but we argue that Dm, n(u) and Tm, n(u) are nonsingular for every in certain cabled link state representations. For generic ?, we derive the fusion hierarchies and the associated T- and Y-systems. For the logarithmic theories, the closure of the fusion hierarchies at n = p? translates into functional relations of polynomial degree p? for Dm, 1(u) and Tm, 1(u). We also derive the closure of the Y-systems for the logarithmic theories. The T- and Y-systems are the key to exact integrability, and we observe that the underlying structure of these functional equations relate to Dynkin diagrams of affine Lie algebras.

First moment of Rankin-Selberg central L-values and subconvexity in the level aspect

Abstract: Let 1≤N

Computing (ℓ,ℓ)-isogenies in polynomial time on Jacobians of genus 2 curves

Abstract: In this paper, we compute l-isogenies between abelian varieties over a field of characteristic different from 2 in polynomial time in l, when l is an odd prime which is coprime to the characteristic. We use level n symmetric theta structure where n = 2 or n = 4. In a second part of this paper we explain how to convert between Mumford coordinates of Jacobians of genus 2 hyperelliptic curves to theta coordinates of level 2 or 4. Combined with the preceding algorithm, this gives a method to compute (l,l)-isogenies in polynomial time on Jacobians of genus 2 curves.

A Generalized Chinese Remainder Theorem for Two Integers

Abstract: A generalized Chinese remainder theorem (CRT) for the determination of two integers is studied in this letter, where the correspondence between the remainders and the two integers in each residue set is not known. A better range than the existing known ones of two integers that can be uniquely determined from their residue sets is first obtained. Then, a closed-form and simple determination algorithm is proposed. Finally, a better sufficient condition on the range of determinable two integers is obtained when the number of erroneous residue sets is given. The study is motivated and has applications in the determination of multiple frequencies from multiple undersampled waveforms.

A refinement of the abc conjecture

Abstract: Based on recent work, by the first and third authors, on the distribution of the squarefree kernel of an integer, we present precise refinements of the famous abc conjecture. These rest on the sole heuristic assumption that, whenever a and b are coprime, then the kernels of a, b and c = a + b are statistically independent.

Classification of global phase portraits of planar quartic quasi-homogeneous polynomial differential systems

Abstract: This paper is devoted to the complete classification of global phase portraits of quasi-homogeneous but non-homogeneous coprime planar polynomial differential systems of degree $$4$$ . To prove our result, we firstly study the canonical forms for these systems. Then, the global topological structures of the systems having canonical forms are studied by using the quasi-homogeneous blow-up technique for the finite singularities and the Poincare-Lyapunov compactification for the infinite singularities. Finally we perform a topological classification for the set of global phase portraits.

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. 10.1017/S0004972713000361

RSOS Quantum Chains Associated with Off-Critical Minimal Models and $\mathbb{Z}_n$ Parafermions

Abstract: We consider the $\varphi_{1,3}$ off-critical perturbation ${\cal M}(m,m';t)$ of the general non-unitary minimal models where $2\le m\le m'$ and $m, m'$ are coprime and $t$ measures the departure from criticality corresponding to the $\varphi_{1,3}$ integrable perturbation. We view these models as the continuum scaling limit in the ferromagnetic Regime III of the Forrester-Baxter Restricted Solid-On-Solid (RSOS) models on the square lattice. We also consider the RSOS models in the antiferromagnetic Regime II related in the continuum scaling limit to $\mathbb{Z}_n$ parfermions with $n=m'-2$. Using an elliptic Yang-Baxter algebra of planar tiles encoding the allowed face configurations, we obtain the Hamiltonians of the associated quantum chains defined as the logarithmic derivative of the transfer matrices with periodic boundary conditions. The transfer matrices and Hamiltonians act on a vector space of paths on the $A_{m'-1}$ Dynkin diagram whose dimension is counted by generalized Fibonacci numbers.

Squarefree values of trinomial discriminants

Abstract: The discriminant of a trinomial of the form $x^n \pm x^m \pm 1$ has the form $\pm n^n \pm (n-m)^{n-m} m^m$ if $n$ and $m$ are relatively prime. We investigate when these discriminants have nontrivial square factors. We explain various unlikely-seeming parametric families of square factors of these discriminant values: for example, when $n$ is congruent to 2 (mod 6) we have that $((n^2-n+1)/3)^2$ always divides $n^n - (n-1)^{n-1}$. In addition, we discover many other square factors of these discriminants that do not fit into these parametric families. The set of primes whose squares can divide these sporadic values as $n$ varies seems to be independent of $m$, and this set can be seen as a generalization of the Wieferich primes, those primes $p$ such that $2^{p-1}$ is congruent to 1 (mod $p^2$). We provide heuristics for the density of squarefree values of these discriminants and the density of these "sporadic" primes.

Error Correction in Polynomial Remainder Codes with Non-Pairwise Coprime Moduli and Robust Chinese Remainder Theorem for Polynomials

Abstract: This paper investigates polynomial remainder codes with non-pairwise coprime moduli. We first consider a robust reconstruction problem for polynomials from erroneous residues when the degrees of all residue errors are assumed small, namely robust Chinese Remainder Theorem (CRT) for polynomials. It basically says that a polynomial can be reconstructed from erroneous residues such that the degree of the reconstruction error is upper bounded by $\tau$ whenever the degrees of all residue errors are upper bounded by $\tau$, where a sufficient condition for $\tau$ and a reconstruction algorithm are obtained. By releasing the constraint that all residue errors have small degrees, another robust reconstruction is then presented when there are multiple unrestricted errors and an arbitrary number of errors with small degrees in the residues. By making full use of redundancy in moduli, we obtain a stronger residue error correction capability in the sense that apart from the number of errors that can be corrected in the previous existing result, some errors with small degrees can be also corrected in the residues. With this newly obtained result, improvements in uncorrected error probability and burst error correction capability in a data transmission are illustrated.

On Solving a Generalized Chinese Remainder Theorem in the Presence of Remainder Errors

Abstract: In estimating frequencies given that the signal waveforms are undersampled multiple times, Xia et. al. proposed to use a generalized version of Chinese remainder Theorem (CRT), where the moduli are $M_1, M_2, \cdots, M_k$ which are not necessarily pairwise coprime. If the errors of the corrupted remainders are within $\tau=\sds \max_{1\le i\le k} \min_{\stackrel{1\le j\le k}{j eq i}} \frac{\gcd(M_i,M_j)}4$, their schemes can be used to construct an approximation of the solution to the generalized CRT with an error smaller than $\tau$. Accurately finding the quotients is a critical ingredient in their approach. In this paper, we shall start with a faithful historical account of the generalized CRT. We then present two treatments of the problem of solving generalized CRT with erroneous remainders. The first treatment follows the route of Wang and Xia to find the quotients, but with a simplified process. The second treatment considers a simplified model of generalized CRT and takes a different approach by working on the corrupted remainders directly. This approach also reveals some useful information about the remainders by inspecting extreme values of the erroneous remainders modulo $4\tau$. Both of our treatments produce efficient algorithms with essentially optimal performance. Finally, this paper constructs a counterexample to prove the sharpness of the error bound $\tau$.

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.

Linear, non-homogeneous, symmetric patterns and prime power generators in numerical semigroups associated to combinatorial configurations

Abstract: We prove that the numerical semigroups associated to the combinatorial configurations satisfy a family of linear, non-homogeneous, symmetric patterns. We use these patterns to prove an upper bound of the conductor and we also give an upper bound of the multiplicity. Also, we compare bounds of the conductor of numerical semigroups associated to balanced configurations, and to configurations with coprime parameters. The proof of the latter involves a bound of the conductor of prime power generated numerical semigroups.

On the sum of dilations of a set

Abstract: We show that for any relatively prime integers $1\leq p

Soft-thresholding for spectrum sensing with coprime samplers

Abstract: Coprime Sampling has been recently proposed to efficiently estimate the spectrum of wideband signals, using sampling rates which can be significantly lower than the Nyquist rate. While the method has been shown to work well when large number of samples are available for estimating the autocorrelation, the effect of fewer samples on the performance of coprime spectrum estimation has not been addressed so far. This paper addresses this issue by employing a denoising scheme on the spectral estimates, as a l_1 norm penalized quadratic program. The solution to this problem results in the so-called soft thresholding operator on the spectral estimates, which inherently promotes sparsity. It also helps to combat the effect of spurious peaks resulting from the finite sample averaging. The probabilities of detecting active and inactive bands are also explicitly characterized and they converge to unity by increasing the number (L) of sub Nyquist samples available to compute the estimates. The effectiveness of the proposed method is demonstrated through numerical examples.

Close to Uniform Prime Number Generation With Fewer Random Bits

Abstract: In this paper, we analyze several variants of a simple method for generating prime numbers with fewer random bits. To generate a prime $p$ less than $x$, the basic idea is to fix a constant $q\propto x^{1-\varepsilon}$, pick a uniformly random $a

Jeśmanowicz' conjecture with fermat numbers

Abstract: Let $a,b,c$ be relatively prime positive integers such that $a^{2}+b^{2}=c^{2}.$ In 1956, Je\'{s}manowicz conjectured that for any positive integer $n$, the only solution of $(an)^{x}+(bn)^{y}=(cn)^{z}$ in positive integers is $(x,y,z)=(2,2,2)$. Let $k\geq 1$ be an integer and $F_k=2^{2^k}+1$ be $k$-th Fermat number. In this paper, we show that Je\'{s}manowicz' conjecture is true for Pythagorean triples $(a,b,c)=(F_k-2,2^{2^{k-1}+1},F_k)$.

The number of simultaneous core partitions

Abstract: Amdeberhan conjectured that the number of $(t,t+1, t+2)$-core partitions is $\sum_{0\leq k\leq [\frac{t}{2}]}\frac{1}{k+1}\binom{t}{2k}\binom{2k}{k}$. In this paper, we obtain the generating function of the numbers $f_t$ of $(t, t + 1, ..., t + p)$-core partitions. In particular, this verifies that Amdeberhan's conjecture is true. We also prove that the number of $(t_1,t_2,..., t_m)$-core partitions is finite if and only if gcd$(t_1,t_2,..., t_m)=1,$ which extends Anderson's result on the finiteness of the number of $(t_1,t_2)$-core partitions for coprime positive integers $t_1$ and $t_2$ and thus rediscover a result of Keith and Nath with a different proof.

On a conjecture of Wilf about the Frobenius number

Abstract: Given coprime positive integers $a_1 < ...< a_d$, the Frobenius number $F$ is the largest integer which is not representable as a non-negative integer combination of the $a_i$. Let $g$ denote the number of all non-representable positive integers: Wilf conjectured that $d \geq \frac{F+1}{F+1-g}$. We prove that for every fixed value of $\lceil \frac{a_1}{d} \rceil$ the conjecture holds for all values of $a_1$ which are sufficiently large and are not divisible by a finite set of primes. We also propose a generalization in the context of one-dimensional local rings and a question on the equality $d = \frac{F+1}{F+1-g}$.