Showing papers on "Coprime integers published in 2015"

Generalized Coprime Array Configurations for Direction-of-Arrival Estimation

Abstract: A coprime array uses two uniform linear subarrays to construct an effective difference coarray with certain desirable characteristics, such as a high number of degrees-of-freedom for direction-of-arrival (DOA) estimation. In this paper, we generalize the coprime array concept with two operations. The first operation is through the compression of the inter-element spacing of one subarray and the resulting structure treats the existing variations of coprime array configurations as well as the nested array structure as its special cases. The second operation exploits two displaced subarrays, and the resulting coprime array structure allows the minimum inter-element spacing to be much larger than the typical half-wavelength requirement, making them useful in applications where a small interelement spacing is infeasible. The performance of the generalized coarray structures is evaluated using their difference coarray equivalence. In particular, we derive the analytical expressions for the coarray aperture, the achievable number of unique lags, and the maximum number of consecutive lags for quantitative evaluation, comparison, and design of coprime arrays. The usefulness of these results is demonstrated using examples applied for DOA estimations utilizing both subspace-based and sparse signal reconstruction techniques.

Remarks on the Spatial Smoothing Step in Coarray MUSIC

Abstract: Sparse arrays such as nested and coprime arrays use a technique called spatial smoothing in order to successfully perform MUSIC in the difference-coarray domain. In this paper it is shown that the spatial smoothing step is not necessary in the sense that the effect achieved by that step can be obtained more directly. In particular, with ${\widetilde {\bf R}_{ss}}$ denoting the spatial smoothed matrix with finite snapshots, it is shown here that the noise eigenspace of this matrix can be directly obtained from another matrix $\widetilde {\bf R}$ which is much easier to compute from data.

Supersparse Linear Integer Models for Optimized Medical Scoring Systems

Abstract: Scoring systems are linear classification models that only require users to add, subtract and multiply a few small numbers in order to make a prediction. These models are in widespread use by the medical community, but are difficult to learn from data because they need to be accurate and sparse, have coprime integer coefficients, and satisfy multiple operational constraints. We present a new method for creating data-driven scoring systems called a Supersparse Linear Integer Model (SLIM). SLIM scoring systems are built by solving an integer program that directly encodes measures of accuracy (the 0-1 loss) and sparsity (the $\ell_0$-seminorm) while restricting coefficients to coprime integers. SLIM can seamlessly incorporate a wide range of operational constraints related to accuracy and sparsity, and can produce highly tailored models without parameter tuning. We provide bounds on the testing and training accuracy of SLIM scoring systems, and present a new data reduction technique that can improve scalability by eliminating a portion of the training data beforehand. Our paper includes results from a collaboration with the Massachusetts General Hospital Sleep Laboratory, where SLIM was used to create a highly tailored scoring system for sleep apnea screening

Schur Indices, BPS Particles, and Argyres-Douglas Theories

Abstract: We conjecture a precise relationship between the Schur limit of the superconformal index of four-dimensional $\mathcal{N}=2$ field theories, which counts local operators, and the spectrum of BPS particles on the Coulomb branch. We verify this conjecture for the special case of free field theories, $\mathcal{N}=2$ QED, and $SU(2)$ gauge theory coupled to fundamental matter. Assuming the validity of our proposal, we compute the Schur index of all Argyres-Douglas theories. Our answers match expectations from the connection of Schur operators with two-dimensional chiral algebras. Based on our results we propose that the chiral algebra of the generalized Argyres-Douglas theory $(A_{k-1},A_{N-1})$ with $k$ and $N$ coprime, is the vacuum sector of the $(k,k+N)$ $W_{k}$ minimal model, and that the Schur index is the associated vacuum character.

Superconformal indices of generalized Argyres-Douglas theories from 2d TQFT

Abstract: We study superconformal indices of 4d N=2 class S theories with certain irregular punctures called type $I_{k, N}$. This class of theories include generalized Argyres-Douglas theories of type $(A_{k-1}, A_{N-1})$ and more. We conjecture the superconformal indices in certain simplified limits based on the TQFT structure of the class S theories by writing an expression for the wave function corresponding to the puncture $I_{k, N}$. We write the Schur limit of the wave function when $k$ and $N$ are coprime. When $k=2$, we also conjecture a closed-form expression for the Hall-Littlewood index and the Macdonald index for odd $N$. From the index, we argue that certain short-multiplet which can appear in the OPE of the stress-energy tensor is absent in the $(A_1, A_{2n})$ theory. We also discuss the mixed Schur indices for the N=1 class S theories with irregular punctures.

Hardware Efficient Mixed Radix-25/16/9 FFT for LTE Systems

Abstract: In this paper, we propose a hardware-efficient mixed generalized high-radix (GHR) reconfigurable fast Fourier transform (FFT) processor for long-term evolution applications. The GHR processor based on radix-25/16/9 uses a 2-D factorization scheme as the high-radix unit and a 1-D factorization method as the system data routing technology. The 2-D factorization scheme is implemented by an enhanced delay element matrix structure, which supports 25-, 16-, 9-, 8-, 5-, 4-, 3-, and 2-point FFTs. Two different designs were implemented. One design (called discrete Fourier transform core) supports 34 different transform sizes from 12 to 1296 points, while the other design (called FFT core) supports five different power-of-two sizes from 128 to 2048 points. The 1-D factorization method is performed by a coprime accessing technology, which accesses the data in parallel without conflict using a RAM. The GHR combines 2-D and 1-D factorization techniques and improves the throughput by a factor of two to four with comparable hardware cost compared with the previous designs. The speed–area ratio of the proposed scheme is nearly two times better than that of previous FFT processors. Application-specified integrated circuit implementation results based on a 0.18- $\mu{\rm m}$ technology are also provided.

Coprime Synthetic Aperture Radar (CopSAR): A New Acquisition Mode for Maritime Surveillance

Abstract: In this paper, we propose a synthetic aperture radar (SAR) technique able, in the case of bright targets over a dark background, to reduce the amount of data to be stored and processed, and, at the same time, to increase the range swath, with no geometric resolution loss. Accordingly, the proposed approach can be usefully employed in ocean monitoring for ship detection. The technique consists of a new SAR acquisition mode and of very simple processing. It is based on the adaptation of the coprime array beamforming concept to the case of SAR systems: two interlaced sequences of pulses are transmitted, with two sub-Nyquist pulse repetition frequencies (PRFs) that are equal to the Nyquist PRF divided by two coprime integer numbers. Each sequence is separately processed via standard SAR processing, and the two final aliased images are combined in a very simple way to cancel aliasing. We call the proposed approach “coprime SAR” (CopSAR). Three different implementations are proposed, and the effectiveness of the CopSAR concept is demonstrated by using both simulated and real SAR data. It turns out that the significant data amount reduction and the range swath increase are only paid with a reduction of the target-to-background ratio and with the presence of a (nonstringent) limit on ship azimuth size.

Generalized Fermat equations: A miscellany

Abstract: This paper is devoted to the generalized Fermat equation xp + yq = zr, where p, q and r are integers, and x, y and z are nonzero coprime integers. We begin by surveying the exponent triples (p, q, r), including a number of infinite families, for which the equation has been solved to date, detailing the techniques involved. In the remainder of the paper, we attempt to solve the remaining infinite families of generalized Fermat equations that appear amenable to current techniques. While the main tools we employ are based upon the modularity of Galois representations (as is indeed true with all previously solved infinite families), in a number of cases we are led via descent to appeal to a rather intricate combination of multi-Frey techniques.

Coprime arrays and samplers for space-time adaptive processing

Abstract: This paper extends the use of coprime arrays and samplers for the case of moving sources. Space-time adaptive processing (STAP) plays an important role in estimating direction-of-arrivals (DOAs) and radial velocities of emitting sources. However, the detection performance is fundamentally limited by the array geometry and the temporal samplers at each sensor. Coprime arrays and coprime samplers offer an enhanced degree of freedom of O(MN) using only O(M + N) physical sensors or samples. In this paper, we propose coprime joint angle-Doppler estimation (coprime JADE), which incorporates both coprime arrays and coprime samplers with the STAP framework. Nonuniform time samples at different sensors can be used to generate a sampled autocorrelation matrix, from which we compute a spatial smoothed matrix. It will be proved that spatial smoothed matrices can be used in the MUSIC algorithm for parameter estimation. With sufficient snapshots, coprime JADE distinguishes O(M 1 N 1 M 2 2 ) independent sources if it corresponds to coprime arrays and coprime samplers with coprime integers (M 1 ; 1 ) and (M 2 ;N 2 ), respectively. It is verified through simulations that coprime JADE resolves the angle-Doppler information better compared to other conventional algorithms.

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+xϵ] is about xϵ/logx. The second says that the number of primes p

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.

The Catalan Case of Armstrong's Conjecture on Simultaneous Core Partitions

Abstract: A beautiful recent conjecture of Armstrong predicts the average size of a partition that is simultaneously an $s$-core and a $t$-core, where $s$ and $t$ are coprime. Our goal is to prove this conjecture when t=s+1. These simultaneous (s,s+1)-core partitions, which are enumerated by Catalan numbers, have average size $\binom{s+1}{3}/2$.

Nonuniform Multiresolution Analysis on Local Fields of Positive Characteristic

Abstract: A generalization of Mallat’s classic theory of multiresolution analysis (MRA) on local fields of positive characteristic was considered by Jiang et al. (J Math Anal Appl 294:523–532, 2004). In this paper, we present a notion of nonuniform MRA on local field \(K\) of positive characteristic. The associated subspace \(V_0\) of \(L^2(K)\) has an orthonormal basis, a collection of translates of the scaling function \(\varphi \) of the form \(\{ \varphi (x-\lambda ) \}_{ \lambda \in \Lambda }\) where \(\Lambda = \{ 0,r/N \}+ \mathcal{Z}, \,N \ge 1\) is an integer and \(r\) is an odd integer such that \(r\) and \(N\) are relatively prime and \(\mathcal{Z}=\{u(n): n\in \mathbb {N}_{0}\}\). We obtain the necessary and sufficient condition for the existence of associated wavelets and present an algorithm for the construction of nonuniform MRA on local fields starting from a low-pass filter \(m_{0}\) with appropriate conditions.

Constructive noncommutative rank computation in deterministic polynomial time over fields of arbitrary characteristics.

Abstract: We extend our techniques developed in our manuscript mentioned in the subtitle to obtain a deterministic polynomial time algorithm for computing the non-commutative rank together with certificates of linear spaces of matrices over sufficiently large base fields. The key new idea is a reduction procedure that keeps the blow-up parameter small, and there are two methods to implement this idea: the first one is a greedy argument that removes certain rows and columns, and the second one is an efficient algorithmic version of a result of Derksen and Makam. Both methods rely crucially on the regularity lemma in the aforementioned manuscript, and in this note we improve that lemma by removing a coprime condition there. arXiv:1508.00690

Lattice points and simultaneous core partitions

Abstract: We observe that for a and b relatively prime, the "abacus construction" identifies the set of simultaneous (a,b)-core partitions with lattice points in a rational simplex. Furthermore, many statistics on (a,b)-cores are piecewise polynomial functions on this simplex. We apply these results to rational Catalan combinatorics. Using Ehrhart theory, we reprove Anderson's theorem that there are (a+b-1)!/a!b! simultaneous (a,b)-cores, and using Euler-Maclaurin theory we prove Armstrong's conjecture that the average size of an (a,b)-core is (a+b+1)(a-1)(b-1)/24. Our methods also give new derivations of analogous formulas for the number and average size of self-conjugate (a,b)-cores. We conjecture a unimodality result for q rational Catalan numbers, and make preliminary investigations in applying these methods to the (q,t)-symmetry and specialization conjectures. We prove these conjectures for low degree terms and when a=3, connecting them to the Catalan hyperplane arrangement and quadratic permutation statistics.

Simultaneous core partitions: parameterizations and sums

Abstract: Fix coprime $s,t\ge1$. We re-prove, without Ehrhart reciprocity, a conjecture of Armstrong (recently verified by Johnson) that the finitely many simultaneous $(s,t)$-cores have average size $\frac{1}{24}(s-1)(t-1)(s+t+1)$, and that the subset of self-conjugate cores has the same average (first shown by Chen--Huang--Wang). We similarly prove a recent conjecture of Fayers that the average weighted by an inverse stabilizer---giving the "expected size of the $t$-core of a random $s$-core"---is $\frac{1}{24}(s-1)(t^2-1)$. We also prove Fayers' conjecture that the analogous self-conjugate average is the same if $t$ is odd, but instead $\frac{1}{24}(s-1)(t^2+2)$ if $t$ is even. In principle, our explicit methods---or implicit variants thereof---extend to averages of arbitrary powers. The main new observation is that the stabilizers appearing in Fayers' conjectures have simple formulas in Johnson's $z$-coordinates parameterization of $(s,t)$-cores. We also observe that the $z$-coordinates extend to parameterize general $t$-cores. As an example application with $t := s+d$, we count the number of $(s,s+d,s+2d)$-cores for coprime $s,d\ge1$, verifying a recent conjecture of Amdeberhan and Leven.

On Singular Moduli for Arbitrary Discriminants

Abstract: Let $d_1$ and $d_2$ be discriminants of distinct quadratic imaginary orders ${{\cal O}}_{d_1}$ and ${{\cal O}}_{d_2}$ and let $J(d_1,d_2)$ denote the product of differences of CM $j$ -invariants with discriminants $d_1$ and $d_2$ . In 1985, Gross and Zagier gave an elegant formula for the factorization of the integer $J(d_1,d_2)$ in the case that $d_1$ and $d_2$ are relatively prime and discriminants of maximal orders. To compute this formula, they first reduce the problem to counting the number of simultaneous embeddings of ${{\cal O}}_{d_1}$ and ${{\cal O}}_{d_2}$ into endomorphism rings of supersingular curves, and then solve this counting problem.Interestingly, this counting problem also appears when computing class polynomials for invariants of genus $2$ curves. However, in this application, one must consider orders ${{\cal O}}_{d_1}$ and ${{\cal O}}_{d_2}$ that are non-maximal. Motivated by the application to genus $2$ curves, we generalize the methods of Gross and Zagier and give a computable formula for $v_{\ell }(J(d_1,d_2))$ for any pair of discriminants $d_1 eq d_2$ and any prime $\ell \gt 2$ . In the case that $d_1$ is squarefree and $d_2$ is the discriminant of any quadratic imaginary order, our formula can be stated in a simple closed form. We also give a conjectural closed formula when the conductors of $d_1$ and $d_2$ are relatively prime.

Every coprime linear group admits a base of size two

Abstract: Let G be a linear group acting on the finite vector space V and assume that (|G|,|V|)=1. In this paper we prove that G has a base size at most two and this estimate is sharp. This generalizes and strengthens several former results concerning base sizes of coprime linear groups. As a direct consequence, we answer a question of I. M. Isaacs in the affirmative. Via large orbits this is related to the k(GV) theorem.

On a conjecture by Wilf about the Frobenius number

Abstract: Given coprime positive integers $$a_1 < \cdots < a_d$$ , 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 $$d \ge \frac{F +1}{F+1-g}$$ . 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 $$d = \frac{F+1}{F+1-g}$$ .

Zariski Topologies for Coprime and Second Submodules

Abstract: Let M be a non-zero module over an associative (not necessarily commutative) ring. In this paper, we investigate the so-called second and coprime submodules of M. Moreover, we topologize the spectrum Specs(M) of second submodules of M and the spectrum Specc(M) of coprime submodules of M, study several properties of these spaces and investigate their interplay with the algebraic properties of M.

Broadband implementation of coprime linear microphone arrays for direction of arrival estimationa)

Abstract: Coprime arrays represent a form of sparse sensing which can achieve narrow beams using relatively few elements, exceeding the spatial Nyquist sampling limit. The purpose of this paper is to expand on and experimentally validate coprime array theory in an acoustic implementation. Two nested sparse uniform linear subarrays with coprime number of elements ( M and N) each produce grating lobes that overlap with one another completely in just one direction. When the subarray outputs are combined it is possible to retain the shared beam while mostly canceling the other superfluous grating lobes. In this way a small number of microphones ( N+M−1) creates a narrow beam at higher frequencies, comparable to a densely populated uniform linear array of MN microphones. In this work beampatterns are simulated for a range of single frequencies, as well as bands of frequencies. Narrowband experimental beampatterns are shown to correspond with simulated results even at frequencies other than the arrays design frequency. N...

p-adic Heights of Heegner points on Shimura curves

Abstract: Let f be a primitive Hilbert modular form of parallel weight 2 and level N for the totally real field F, and let p be a rational prime coprime to 2N. If f is ordinary at p and E is a CM extension of F of relative discriminant Δ prime to Np, we give an explicit construction of the p-adic Rankin–Selberg L-function Lp(fE,⋅). When the sign of its functional equation is − 1, we show, under the assumption that all primes ℘∣p are principal ideals of OF that split in OE, that its central derivative is given by the p-adic height of a Heegner point on the abelian variety A associated with f. This p-adic Gross–Zagier formula generalises the result obtained by Perrin-Riou when F = ℚ and (N,E) satisfies the so-called Heegner condition. We deduce applications to both the p-adic and the classical Birch and Swinnerton-Dyer conjectures for A.

Vector-valued nonuniform multiresolution analysis on local fields

Abstract: A multiresolution analysis (MRA) on local fields of positive characteristic was defined by Shah and Abdullah for which the translation set is a discrete set which is not a group. In this paper, we continue the study based on this nonstandard setting and introduce vector-valued nonuniform multiresolution analysis (VNUMRA) where the associated subspace V0 of L2(K, ℂM) has an orthonormal basis of the form {Φ (x - λ)}λ∈Λ where Λ = {0, r/N} + 𝒵, N ≥ 1 is an integer and r is an odd integer such that r and N are relatively prime and 𝒵 = {u(n) : n ∈ ℕ0}. We establish a necessary and sufficient condition for the existence of associated wavelets and derive an algorithm for the construction of VNUMRA on local fields starting from a vector refinement mask G(ξ) with appropriate conditions. Further, these results also hold for Cantor and Vilenkin groups.

Basis Construction for Range Estimation by Phase Unwrapping

Abstract: We consider the problem of estimating the distance, or range, between two locations by measuring the phase of a sinusoidal signal transmitted between the locations. This method is only capable of unambiguously measuring range within an interval of length equal to the wavelength of the signal. To address this problem signals of multiple different wavelengths can be transmitted. The range can then be measured within an interval of length equal to the least common multiple of these wavelengths. Estimation of the range requires solution of a problem from computational number theory called the closest lattice point problem. Algorithms to solve this problem require a basis for this lattice. Constructing a basis is non-trivial and an explicit construction has only been given in the case that the wavelengths can be scaled to pairwise relatively prime integers. In this paper we present an explicit construction of a basis without this assumption on the wavelengths. This is important because the accuracy of the range estimator depends upon the wavelengths. Simulations indicate that significant improvement in accuracy can be achieved by using wavelengths that cannot be scaled to pairwise relatively prime integers.

Consecutive primes in tuples

Abstract: In a stunning new advance towards the Prime k-tuple Conjecture, Maynard and Tao have shown that if k is sufficiently large in terms ofm, then for any admissible k-tuple H(x) = {gjx + hj} k=1 of linear forms in Z(x) the set H(n) = {gjn + hj} k=1 contains at least m primes for infinitely many n ∈ N. In this short note, we show how the Maynard-Tao theorem may be used to produce m-tuples H(x) for which the m primes that occur in H(n) are consecutive. We answer an old question of Erdýos and Turan by producing strings of m + 1 consecutive primes whose successive gaps �1,...,�m form an increasing (resp. decreasing) sequence. We also show that such strings exist for whichj 1 | �j for 2 6 j 6 m. For any coprime integers a and D > 1 we find strings of consecutive primes of arbitrary length in the congruence class a mod D.

On a conjecture of Gluck

Abstract: Let $$\mathbf {F}(G)$$ and $$b(G)$$ respectively denote the Fitting subgroup and the largest degree of an irreducible complex character of a finite group $$G$$ A well-known conjecture of D Gluck claims that if $$G$$ is solvable then $$|G:\mathbf {F}(G)|\le b(G)^{2}$$ We confirm this conjecture in the case where $$|\mathbf {F}(G)|$$ is coprime to 6 We also extend the problem to arbitrary finite groups and prove several results showing that the largest irreducible character degree of a finite group strongly controls the group structure

Experimental validation of a coprime linear microphone array for high-resolution direction-of-arrival measurementsa)

Abstract: Coprime linear microphone arrays allow for narrower beams with fewer sensors. A coprime microphone array consists of two staggered uniform linear subarrays with M and N microphones, where M and N are coprime with each other. By applying spatial filtering to both subarrays and combining their outputs, M+N-1 microphones yield M⋅N directional bands. In this work, the coprime sampling theory is implemented in the form of a linear microphone array of 16 elements with coprime numbers of 9 and 8. This coprime microphone array is experimentally tested to validate the coprime array theory. Both predicted and measured results are discussed. Experimental results confirm that narrow beampatterns as predicted by the coprime sampling theory can be obtained by the coprime microphone array.

Commutators of elements of coprime orders in finite groups

Abstract: This paper is an attempt to find out which properties of a finite group G can be expressed in terms of commutators of elements of coprime orders. A criterion of solubility of G in terms of such commutators is obtained. We also conjecture that every element of a nonabelian simple group is a commutator of elements of coprime orders and we confirm this conjecture for the alternating groups.

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, the robust Chinese Remainder Theorem (CRT) for polynomials. It basically says that a polynomial can be reconstructed from its 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 relaxing 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. We finally 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 data transmission are illustrated.