Showing papers on "Coprime integers published in 2016"

Super Nested Arrays: Linear Sparse Arrays With Reduced Mutual Coupling—Part I: Fundamentals

Abstract: In array processing, mutual coupling between sensors has an adverse effect on the estimation of parameters (e.g., DOA). While there are methods to counteract this through appropriate modeling and calibration, they are usually computationally expensive, and sensitive to model mismatch. On the other hand, sparse arrays, such as nested arrays, coprime arrays, and minimum redundancy arrays (MRAs), have reduced mutual coupling compared to uniform linear arrays (ULAs). With $N$ denoting the number of sensors, these sparse arrays offer $O({N}^{2})$ freedoms for source estimation because their difference coarrays have $O({N}^{2})$ -long ULA segments. But these well-known sparse arrays have disadvantages: MRAs do not have simple closed-form expressions for the array geometry; coprime arrays have holes in the coarray; and nested arrays contain a dense ULA in the physical array, resulting in significantly higher mutual coupling than coprime arrays and MRAs. This paper introduces a new array called the super nested array, which has all the good properties of the nested array, and at the same time achieves reduced mutual coupling. There is a systematic procedure to determine sensor locations. For fixed $N$ , the super nested array has the same physical aperture, and the same hole-free coarray as does the nested array. But the number of sensor pairs with small separations ( $\lambda /2,2\times \lambda /2$ , etc.) is significantly reduced. Many theoretical properties are proved and simulations are included to demonstrate the superior performance of these arrays. In the companion paper, a further extension called the $Q$ th-order super nested array is developed, which further reduces mutual coupling.

Super Nested Arrays: Linear Sparse Arrays With Reduced Mutual Coupling—Part II: High-Order Extensions

Abstract: In array processing, mutual coupling between sensors has an adverse effect on the estimation of parameters (e.g., DOA). Sparse arrays such as nested arrays, coprime arrays, and minimum redundancy arrays (MRA) have reduced mutual coupling compared to uniform linear arrays (ULAs). These arrays also have a difference coarray with $O\left({N}^{2}\right)$ virtual elements, where $N$ is the number of physical sensors, and can therefore resolve $O\left({N}^{2}\right)$ uncorrelated source directions. But these well-known sparse arrays have disadvantages: MRAs do not have simple closed-form expressions for the array geometry; coprime arrays have holes in the coarray; and nested arrays contain a dense ULA in the physical array, resulting in significantly higher mutual coupling than coprime arrays and MRAs. In a companion paper, a sparse array configuration called the (second-order) super nested array was introduced, which has many of the advantages of these sparse arrays, while removing most of the disadvantages. Namely, the sensor locations are readily computed for any $N$ (unlike MRAs), and the difference coarray is exactly that of a nested array, and therefore hole-free. At the same time, the mutual coupling is reduced significantly (unlike nested arrays). In this paper, a generalization of super nested arrays is introduced, called the $Q$ th-order super nested array. This has all the properties of the second-order super nested array with the additional advantage that mutual coupling effects are further reduced for $Q > 2$ . Many theoretical properties are proved and simulations are included to demonstrate the superior performance of these arrays.

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 using an integer programming problem that directly encodes measures of accuracy (the 0---1 loss) and sparsity (the $$\ell _0$$l0-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 acceptable models without parameter tuning because of the direct control provided over these quantities. 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 is being 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.

Coprime coarray interpolation for DOA estimation via nuclear norm minimization

Abstract: Coprime arrays, consisting of two uniform linear arrays whose inter-element separations are coprime, can resolve O(MN) sources using only O(M + N) sensors. However, holes in the coarray prevent us from using the full coarray in the MUSIC algorithm for DOA estimation. Through interpolation, it may be possible to use the remaining elements of the coarray to increase the degrees of freedom beyond what is captured in the contiguous ULA section in the coarray. Techniques like positive definite Toeplitz completion, array interpolation, and sparse recovery, manage to include all the information in the coarray, but they demand extra fine-tuned parameters and have individual drawbacks. In this paper, a simple and tractable convex framework via nuclear norm minimization is presented. This approach has no extra tuning parameters and overcomes several undesired issues of other techniques. Numerical examples indicate that, in many instances, the proposed method not only increases the estimation accuracy but also distinguishes more sources than other methods.1

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

Abstract: We study superconformal indices of 4d $$ \mathcal{N}=2 $$ 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. Fromtheindex,wearguethatcertainshort-multipletwhichcanappearintheOPEof the stress-energy tensor is absent in the (A 1 , A 2n ) theory. We also discuss the mixed Schur indices for the $$ \mathcal{N}=1 $$ class $$ \mathcal{S} $$ theories with irregular punctures.

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 Catalan number $${\frac{1}{n+1} \left( \begin{array}{ll} 2n \\ n \end{array} \right)}$$ . In this paper, we will generalize this setup to “rational” parking functions indexed by a pair (a, b) of coprime positive integers. These parking functions, which are counted by b a−1, carry a permutation representation of S a in which the number of orbits is the “rational” Catalan number $${\frac{1}{a+b} \left( \begin{array}{ll} a+b \\ a \end{array} \right)}$$ . First, we compute the Frobenius characteristic of the S a -module of (a, b)-parking functions, giving explicit expansions of this symmetric function in the complete homogeneous basis, the power-sum basis, and the Schur basis. Second, we study q-analogues of the rational Catalan numbers, conjecturing new combinatorial formulas for the rational q-Catalan numbers $${\frac{1}{[a+b]_{q}} {{\left[ \begin{array}{ll} a+b \\ a \end{array} \right]}_{q}}}$$ and for the q-binomial coefficients $${{{\left[ \begin{array}{ll} n \\ k \end{array} \right]}_{q}}}$$ . We give a bijective explanation of the division by [a+b] q that proves the equivalence of these two conjectures. Third, we present combinatorial definitions for q, t-analogues of rational Catalan numbers and parking functions, generalizing the Shuffle Conjecture for the classical case. We present several conjectures regarding the joint symmetry and t = 1/q specializations of these polynomials. An appendix computes these polynomials explicitly for small values of a and b.

Construction of de Bruijn Sequences From LFSRs With Reducible Characteristic Polynomials

Abstract: In this paper, a family of new de Bruijn sequences is proposed through the construction of maximum-length nonlinear feedback shift registers (NFSRs). Let $k$ be a positive integer and $p_{0}(x), p_{1}(x), \ldots , p_{k}(x)$ be the primitive polynomials in $\mathbb {F}_{2}[x]$ with their degrees strictly increasing and pairwise coprime. We determine the cycle structure and adjacency graphs of linear feedback shift registers (LFSRs) with characteristic polynomial $q(x)=\prod olimits _{i=0}^{k}p_{i}(x)$ . In the case that $p_{0}(x)=1+x$ , an algorithm is proposed to produce maximum-length NFSRs from these LFSRs, and it is shown that the algorithm can generate $O(2^{(2^{k}-1)n})~n$ -stage maximum-length NFSRs with memory complexity $O(2^{k}kn)$ and time complexity $O(2^{n-d_{k}}kn)$ , where $n$ and $d_{k}$ are the degrees of $q(x)$ and $p_{k}(x)$ , respectively. Finally, we illustrate the proposed algorithm in the case of $k=2$ . In this case, we prove that for any integer $n\geq 8$ , the algorithm can produce $n$ -stage maximum-length NFSRs with time complexity as low as $O(n^{{\rm {log}{log}}(n)}$ ).

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.

On constacyclic codes over finite fields

Abstract: Constacyclic codes are generalizations of cyclic codes, which form a well-known family of linear codes containing many optimal codes. In this paper, we determine all constacyclic codes of length lpw over the finite field źźq$\mathbb {F}_{q}$ with q elements, where q is a power of the prime p, l is a positive integer coprime to q and w ź 0 is an integer. We also illustrate our results with some examples.

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.

Robust adaptive beamforming based on DOA support using decomposed coprime subarrays

Abstract: In this paper, we propose a novel robust adaptive beamforming algorithm with direction-of-arrival (DOA) support for the coprirne array. Specifically, by using the property of coprirne number, we may estimate the DOAs of sources by matching two super-resolution spatial spectra of the pair of decomposed coprime subarrays. After that, the power of each source can be estimated via a covariance matrix joint estimation problem corresponding to the pair of decomposed coprime sub-arrays. Taking the estimated DOAs and their corresponding power as the support information, the interference-plus-noise covariance matrix for the coprime array can be reconstructed, from which the minimum variance distortionless response beamformer weight vector can be calculated. Simulation results show that the proposed adaptive beamforming algorithm is more robust to signal look direction mismatch than the existing algorithms.

Multi-target localization using frequency diverse coprime arrays with coprime frequency offsets

Abstract: The performance of the frequency diverse array (FDA) radar is fundamentally limited by the geometry of the array and the frequency offset. In this paper, we overcome this limitation by introducing a novel sparsity-based multi-target localization approach incorporating both coprime array and coprime frequency offset. The covariance matrix of the received signals corresponding to all sensors and employed frequencies is formulated to generate a space-frequency virtual difference coarrays. The proposed approach enables the localization of up to O(M2 N2) targets using O(M + N) physical sensors with O(M + N) frequencies for a coprime pair of M and N. The joint DOA and range estimation is cast as a sparse reconstruction problem and solved using the complex multi-task Bayesian compressive sensing technique.

The Generalized Fermat Equation

Abstract: We survey approaches to solving the generalized Fermat equation $$\displaystyle{x^{p} + y^{q} = z^{r}}$$ in relatively prime integers x, y and z, and integers p, q and r ≥ 2.

Low-lying zeros of elliptic curve L-functions: Beyond the Ratios Conjecture

Abstract: We study the low-lying zeros of L-functions attached to quadratic twists of a given elliptic curve E defined over . We are primarily interested in the family of all twists coprime to the conductor of E and compute a very precise expression for the corresponding 1-level density. In particular, for test functions whose Fourier transforms have sufficiently restricted support, we are able to compute the 1-level density up to an error term that is significantly sharper than the square-root error term predicted by the L-functions Ratios Conjecture.

Two-dimensional DOA estimation using parallel coprime subarrays

Abstract: A conventional coprime array is a linear array, which consists of two uniform linear subarrays to construct an effective difference coarray with certain desirable characteristics. In this paper, we propose a parallel coprime array structure and a novel algorithm for two-dimensional (2-D) direction-of-arrival (DOA) estimation. By vectorizing the cross-covariance matrix of subarray data, the resulting virtual difference coarray enables resolving more signals than the number of antennas. The 2-D DOA estimation problem is cast as two separate one-dimensional DOA estimation problems, where the estimated azimuth and elevation angles can be properly associated. Compared with other methods, such as, the propagator method (PM) and the rank-reduction (RARE) based algorithms, the proposed method resolves more signals and achieves improved estimation performance.

A sufficient condition for nilpotency of the commutator subgroup

Abstract: Let G be a finite group with the property that if a and b are commutators of coprime orders, then |ab| = |a||b|. We show that G′ is nilpotent.

DOA Estimation of Coherent Signals on Coprime Arrays Exploiting Fourth-Order Cumulants

Abstract: This paper considers the problem of direction-of-arrival (DOA) estimation of coherent signals on passive coprime arrays, where we resort to the fourth-order cumulants of the received signal to explore more information. A fourth-order cumulant matrix (FCM) is introduced for the coprime arrays. The special structure of the FCM is combined with the array configuration to resolve the coherent signals. Since each sparse array of the coprime arrays is uniform, a series of overlapping identical subarrays can be extracted. Using this property, we propose a generalized spatial smoothing scheme applied to the FCM. From the smoothed FCM, the DOAs of both the coherent and independent signals can be successfully estimated on the pseudo-spectrum generated by the fourth-order MUSIC algorithm. To overcome the problem of occasional false peak appearing on the pseudo-spectrum, we use a supplementary sparse array whose inter-sensor spacing is coprime to that of either existing sparse array. From the combined spectrum aided by the supplementary sensors, the false peaks are removed while the true peaks remain. The effectiveness of the proposed methods is demonstrated by simulation examples.

Norms of roots of trinomials

Abstract: The behavior of norms of roots of univariate trinomials $$z^{s+t} + p z^t + q \in \mathbb {C}[z]$$ for fixed support $$A = \{0,t,s+t\} \subset \mathbb {N}$$ with respect to the choice of coefficients $$p,q \in \mathbb {C}$$ is a classical late 19th and early 20th century problem. Although algebraically characterized by P. Bohl in 1908, the geometry and topology of the corresponding parameter space of coefficients had yet to be revealed. Assuming s and t to be coprime we provide such a characterization for the space of trinomials by reinterpreting the problem in terms of amoeba theory. The roots of given norm are parameterized in terms of a hypotrochoid curve along a $$\mathbb {C}$$ -slice of the space of trinomials, with multiple roots of this norm appearing exactly on the singularities. As a main result, we show that the set of all trinomials with support A and certain roots of identical norm, as well as its complement can be deformation retracted to the torus knot $$K(s+t,s)$$ , and thus are connected but not simply connected. An exception is the case where the t-th smallest norm coincides with the $$(t+1)$$ -st smallest norm. Here, the complement has a different topology since it has fundamental group $$\mathbb {Z}^2$$ .

Floer field theory for coprime rank and degree

Abstract: We construct partial category-valued field theories in (2+1)-dimensions using Lagrangian Floer theory in moduli spaces of central-curvature unitary connections with fixed determinant of rank r and degree d where r,d are coprime positive integers. These theories associate to a closed, connected, oriented surface the Fukaya category of the moduli space, and to a connected bordism between two surfaces a functor between the Fukaya categories. We obtain the latter by combining Cerf theory with holomorphic quilt invariants.

Cyclic Sieving and Rational Catalan Theory

Abstract: Let $a < b$ be coprime positive integers. Armstrong, Rhoades, and Williams (2013) defined a set NC(a,b) of `rational noncrossing partitions', which form a subset of the ordinary noncrossing partitions of $\{1, 2, \dots, b-1\}$. Confirming a conjecture of Armstrong et. al., we prove that NC(a,b) is closed under rotation and prove an instance of the cyclic sieving phenomenon for this rotational action. We also define a rational generalization of the $\mathfrak{S}_a$-noncrossing parking functions of Armstrong, Reiner, and Rhoades.

A note on a result of Guo and Isaacs about p-supersolubility of finite groups

Abstract: In this note, global information about a finite group is obtained by assuming that certain subgroups of some given order are S-semipermutable. Recall that a subgroup H of a finite group G is said to be S-semipermutable if H permutes with all Sylow subgroups of G of order coprime to \({\lvert H\rvert}\). We prove that for a fixed prime p, a given Sylow p-subgroup P of a finite group G, and a power d of p dividing \({\lvert G\rvert}\) such that \({1\le d d}\). This extends the main result of Guo and Isaacs in (Arch. Math. 105:215–222 2015). We derive some theorems that extend some known results concerning S-semipermutable subgroups.

Beyond double transitivity: Capacity-achieving cyclic codes on erasure channels

Abstract: Recently, sequences of error-correcting codes with doubly-transitive permutation groups were shown to achieve capacity on erasure channels under symbol-wise maximum a posteriori (MAP) decoding From this, it follows that Reed-Muller and primitive narrow-sense BCH codes achieve capacity in the same setting In this article, we extend this result to a large family of cyclic codes by considering codes whose permutation groups satisfy a condition weaker than double transitivity The article combines two simple technical contributions First, we show that the transition width of a monotone boolean function is O(1/log k), where k is the size of the smallest orbit induced by its symmetry group The proof is based on Talagrand's lower bound on influences for monotone boolean functions Second, we consider the extrinsic information transfer (EXIT) function of an Fq-linear cyclic code whose blocklength N divides qt − 1 and is coprime with q − 1 We show that this EXIT function is a monotone boolean function whose symmetry group contains no orbits of size smaller than the smallest prime divisor of t Combining these, we show that sequences of cyclic codes, whose blocklengths satisfy the above conditions, achieve capacity on the q-ary erasure channel if all prime divisors of t tend to infinity

Three-level prime arrays for sparse sampling in direction of arrival estimation

Abstract: This paper proposes array configurations that can be used for sparse direction-of-arrival (DOA) estimation. The proposed array uses three uniform linear subarrays where the number of elements in the subarrays are taken to be pairwise coprime integers. The proposed array is referred to as a three-level prime array (3LPA). If the number of elements are pairwise coprime integers and primitive Pythagorean triple (PPT), a special case of the 3LPA is generated and we refer to as a Pythagorean array (PA). The elements of subarray 1 are spaced by the number of elements of subarray 2 or the number of elements of subarray 3. The same is done for subarray 2 and subarray 3 such that the subarrays share only their first antenna element. The main objective of the proposed arrays is to increase the degree-of-freedom (DOF) using a small aperture size. To handle this, the DOF is optimized and formulated as a function of the number of elements in the subarrays. For the same number of elements, the proposed array has smaller aperture and achieve more unique lags and consecutive lags and consequently large DOF compared with coprime array. Simulation results confirm the advantage of the proposed configurations compared to prototype coprime arrays.

Finite groups and degrees of irreducible monomial characters

Abstract: Let G be a finite solvable group, let Irrm(G) be the set of all irreducible monomial characters of G and let p be a prime. We prove that if p|χ(1) for every nonlinear χ ∈Irrm(G), then G has a normal p-complement, and if p is relatively prime to χ(1) for every χ ∈Irrm(G), then G has a normal Sylow p-subgroup.

Super-approximation, II: the p-adic and bounded power of square-free integers cases

Abstract: Let $\Omega$ be a finite symmetric subset of GL$_n(\mathbb{Z}[1/q_0])$, and $\Gamma:=\langle \Omega \rangle$. Then the family of Cayley graphs $\{{\rm Cay}(\pi_m(\Gamma),\pi_m(\Omega))\}_m$ is a family of expanders as $m$ ranges over fixed powers of square-free integers and powers of primes that are coprime to $q_0$ if and only if the connected component of the Zariski-closure of $\Gamma$ is perfect. Some of the immediate applications, e.g. orbit equivalence rigidity, {\em largeness} of certain $\ell$-adic Galois representations, are also discussed.