Common Zeros Research Articles

CNF Satisfiability in a Subspace and Related Problems

We introduce the problem of finding a satisfying assignment to a CNF formula that must further belong to a prescribed input subspace. Equivalent formulations of the problem include finding a point outside a union of subspaces (the Union-of-Subspace Avoidance (USA) problem), and finding a common zero of a system of polynomials over \({\mathbb {F}}_2\) each of which is a product of affine forms. We focus on the case of k-CNF formulas (the \({k}-\textsc {Sub}-\textsc {Sat}\) problem). Clearly, \({k}-\textsc {Sub}-\textsc {Sat}\) is no easier than k-SAT, and might be harder. Indeed, via simple reductions we show that \({2}-\textsc {Sub}-\textsc {Sat}\) is NP-hard, and \({\small \mathrm {W}}[1]\)-hard when parameterized by the co-dimension of the subspace. We also prove that the optimization version Max-\({2}-\textsc {Sub}-\textsc {Sat}\) is NP-hard to approximate better than the trivial 3/4 ratio even on satisfiable instances. On the algorithmic front, we investigate fast exponential algorithms which give non-trivial savings over brute-force algorithms. We give a simple branching algorithm with running time \(O^*(1.5)^r\) for \({2}-\textsc {Sub}-\textsc {Sat}\), where r is the subspace dimension, as well as an \(O^*(1.4312)^n\) time algorithm where n is the number of variables. Turning to \({k}-\textsc {Sub}-\textsc {Sat}\) for \(k \geqslant 3\), while known algorithms for solving a system of degree k polynomial equations already imply a solution with running time \(\approx 2^{r(1-1/2k)}\), we explore a more combinatorial approach. Based on an analysis of critical variables (a key notion underlying the randomized k-SAT algorithm of Paturi, Pudlak, and Zane), we give an algorithm with running time \(\approx {n\atopwithdelims (){\leqslant t}} 2^{n-n/k}\) where n is the number of variables and t is the co-dimension of the subspace. This improves upon the running time of the polynomial equations approach for small co-dimension. Our combinatorial approach also achieves polynomial space in contrast to the algebraic approach that uses exponential space. We also give a PPZ-style algorithm for \({k}-\textsc {Sub}-\textsc {Sat}\) with running time \(\approx 2^{n-n/2k}\). This algorithm is in fact oblivious to the structure of the subspace, and extends when the subspace-membership constraint is replaced by any constraint for which partial satisfying assignments can be efficiently completed to a full satisfying assignment. Finally, for systems of O(n) polynomial equations in n variables over \({\mathbb {F}}_2\), we give a fast exponential algorithm when each polynomial has bounded degree irreducible factors (but can otherwise have large degree) using a degree reduction trick.

Approximating a Solution of the Inclusion Problem for an Infinite Family of Monotone Operators in Hadamard Spaces and Its Applications

In this paper, a solution to the inclusion problem for an infinite family of monotone operators in Hadamard spaces is approximated. Strong convergence and Δ-convergence to a common zero of an infinite family of monotone operators are established. To support these results, some applications in convex optimization and fixed point theory are also presented.

Joint learning of multiple Granger causal networks via non-convex regularizations: Inference of group-level brain connectivity

This paper considers joint learning of multiple sparse Granger graphical models to discover underlying common and differential Granger causality (GC) structures across multiple time series. This can be applied to drawing group-level brain connectivity inferences from a homogeneous group of subjects or discovering network differences among groups of signals collected under heterogeneous conditions. By recognizing that the GC of a single multivariate time series can be characterized by common zeros of vector autoregressive (VAR) lag coefficients, a group sparse prior is included in joint regularized least-squares estimations of multiple VAR models. Group-norm regularizations based on group- and fused-lasso penalties encourage a decomposition of multiple networks into a common GC structure, with other remaining parts defined in individual-specific networks. Prior information about sparseness and sparsity patterns of desired GC networks are incorporated as relative weights, while a non-convex group norm in the penalty is proposed to enhance the accuracy of network estimation in low-sample settings. Extensive numerical results on simulations illustrated our method’s improvements over existing sparse estimation approaches on GC network sparsity recovery. Our methods were also applied to available resting-state fMRI time series from the ADHD-200 data sets to learn the differences of causality mechanisms, called effective brain connectivity, between adolescents with ADHD and typically developing children. Our analysis revealed that parts of the causality differences between the two groups often resided in the orbitofrontal region and areas associated with the limbic system, which agreed with clinical findings and data-driven results in previous studies.

The quasi-algebraic ring of conditions of

An exponential sum is a linear combination of characters of the additive group of . We regard as an analogue of the torus , exponential sums as analogues of Laurent polynomials, and exponential analytic sets (-sets), that is, the sets of common zeros of finite systems of exponential sums, as analogues of algebraic subvarieties of the torus. Using these analogies, we define the intersection number of -sets and apply the De Concini–Procesi algorithm to construct the ring of conditions of the corresponding intersection theory. To construct the intersection number and the ring of conditions, we associate an algebraic subvariety of a multidimensional complex torus with every -set and use the methods of tropical geometry. By computing the intersection number of the divisors of arbitrary exponential sums , we arrive at a formula for the density of the -set of common zeros of the perturbed system , where the perturbation belongs to a set of relatively full measure in . This formula is analogous to the formula for the number of common zeros of Laurent polynomials.

On a Formula for Calculating the Resultant of Two Entire Functions

Using Newton’s recurrent formulae, we find the product of values of an entire function of one variable in zeroes of another entire function. This allows to answer whether they have common zeroes. By that, we propose an approach to construction of the resultant of two entire functions. We also give example illustrating the main result

Upper Bound for Lebesgue Constant of Bivariate Lagrange Interpolation Polynomial on the Second Kind Chebyshev Points

In the paper, we study the upper bound estimation of the Lebesgue constant of the bivariate Lagrange interpolation polynomial based on the common zeros of product Chebyshev polynomials of the second kind on the square [−1,1]2. And, we prove that the growth order of the Lebesgue constant is O((n + 2)2). This result is different from the Lebesgue constant of Lagrange interpolation polynomial on the unit disk, the growth order of which is . And, it is different from the Lebesgue constant of the Lagrange interpolation polynomial based on the common zeros of product Chebyshev polynomials of the first kind on the square [−1,1]2, the growth order of which is O((lnn)2).

The SALT-Readout ASIC for Silicon Strip Sensors of Upstream Tracker in the Upgraded LHCb Experiment.

SALT, a new dedicated readout Application Specific Integrated Circuit (ASIC) for the Upstream Tracker, a new silicon detector in the Large Hadron Collider beauty (LHCb) experiment, has been designed and developed. It is a 128-channel chip using an innovative architecture comprising a low-power analogue front-end with fast pulse shaping and a 40 MSps 6-bit Analog-to-Digital Converter (ADC) in each channel, followed by a Digital Signal Processing (DSP) block performing pedestal and Mean Common Mode (MCM) subtraction and zero suppression. The prototypes of SALT were fabricated and tested, confirming the full chip functionality and fulfilling the specifications. A signal-to-noise ratio of about 20 is achieved for a silicon sensor with a 12 pF input capacitance. In this paper, the SALT architecture and measurements of the chip performance are presented.

The geometry of Hermitian self-orthogonal codes

We prove that if n >k^2 then a k-dimensional linear code of length n over {mathbb F}_{q^2} has a truncation which is linearly equivalent to a Hermitian self-orthogonal linear code. In the contrary case we prove that truncations of linear codes to codes equivalent to Hermitian self-orthogonal linear codes occur when the columns of a generator matrix of the code do not impose independent conditions on the space of Hermitian forms. In the case that there are more than n common zeros to the set of Hermitian forms which are zero on the columns of a generator matrix of the code, the additional zeros give the extension of the code to a code that has a truncation which is equivalent to a Hermitian self-orthogonal code.

Asymptotic vectors of entire curves

We introduce a concept of asymptotic vector of an entire curve with linearly independent components and without common zeros and investigate a relationship between the asymptotic vectors and the Picard exceptional vectors.
 A non-zero vector $\vec{a}=(a_1,a_2,\ldots,a_p)\in \mathbb{C}^{p}$ is called an asymptotic vector for the entire curve $\vec{G}(z)=(g_1(z),g_2(z),\ldots,g_p(z))$ if there exists a continuous curve $L: \mathbb{R}_+\to \mathbb{C}$ given by an equation $z=z\left(t\right)$, $0\le t<\infty $, $\left|z\left(t\right)\right|<\infty $, $z\left(t\right)\to \infty $ as $t\to \infty $ such that$$\lim\limits_{\stackrel{z\to\infty}{z\in L}} \frac{\vec{G}(z)\vec{a} }{\big\|\vec{G}(z)\big\|}=\lim\limits_{t\to\infty} \frac{\vec{G}(z(t))\vec{a} }{\big\|\vec{G}(z(t))\big\|} =0,$$ where $\big\|\vec{G}(z)\big\|=\big(|g_1(z)|^2+\ldots +|g_p(z)|^2\big)^{1/2}$, $\vec{G}(z)\vec{a}=g_1(z)\cdot\bar{a}_1+g_2(z)\cdot\bar{a}_2+\ldots+g_p(z)\cdot\bar{a}_p$. A non-zero vector $\vec{a}=(a_1,a_2,\ldots,a_p)\in \mathbb{C}^{p}$ is called a Picard exceptional vector of an entire curve $\vec{G}(z)$ if the function $\vec{G}(z)\vec{a}$ has a finite number of zeros in $\left\{\left|z\right|<\infty \right\}$.
 We prove that any Picard exceptional vector of transcendental entire curve with linearly independent com\-po\-nents and without common zeros is an asymptotic vector.Here we de\-mon\-stra\-te that the exceptional vectors in the sense of Borel or Nevanlina and, moreover, in the sense of Valiron do not have to be asymptotic. For this goal we use an example of meromorphic function of finite positive order, for which $\infty $ is no asymptotic value, but it is the Nevanlinna exceptional value. This function is constructed in known Goldberg and Ostrovskii's monograph``Value Distribution of Meromorphic Functions''.Other our result describes sufficient conditions providing that some vectors are asymptotic for transcendental entire curve of finite order with linearly independent components and without common zeros. In this result, we require that the order of the Nevanlinna counting function for this curve and for each such a vector is less than order of the curve.At the end of paper we formulate three unsolved problems concerning asymptotic vectors of entire curve.

A Strong Convergence Theorem for a Finite Family of Bregman Demimetric Mappings in a Banach Space under a New Shrinking Projection Method

In this paper, using a new shrinking projection method and new generalizedk-demimetric mappings, we consider the strong convergence for finding a common point of the sets of zero points of maximal monotone mappings, common fixed points of a finite family of Bregmank-demimetric mappings, and common zero points of a finite family of Bregman inverse strongly monotone mappings in a reflexive Banach space. To the best of our knowledge, such a theorem for Bregmank-demimetric mapping is the first of its kind in a Banach space. This manuscript is online on arXiv by the link http://arxiv.org/abs/2107.13254.

Obstructions to the existence of compact Clifford–Klein forms for tangential symmetric spaces

For a homogeneous space [Formula: see text] of reductive type, we consider the tangential homogeneous space [Formula: see text]. In this paper, we give obstructions to the existence of compact Clifford–Klein forms for such tangential symmetric spaces and obtain new tangential symmetric spaces which do not admit compact Clifford–Klein forms. As a result, in the class of irreducible classical semisimple symmetric spaces, we have only two types of symmetric spaces which are not proved not to admit compact Clifford–Klein forms. The existence problem of compact Clifford–Klein forms for homogeneous spaces of reductive type, which was initiated by Kobayashi in 1980s, has been studied by various methods but is not completely solved yet. On the other hand, the one for tangential homogeneous spaces has been studied since 2000s and an analogous criterion was proved by Kobayashi and Yoshino. In concrete examples, further works are needed to verify Kobayashi–Yoshino’s condition by direct calculations. In this paper, some easy-to-check necessary conditions ([Formula: see text][Formula: see text]obstructions) for the existence of compact quotients in the tangential setting are given, and they are applied to the case of symmetric spaces. The conditions are related to various fields of mathematics such as associated pair of symmetric space, Calabi–Markus phenomenon, trivializability of vector bundle (parallelizability, Pontrjagin class), Hurwitz–Radon number and Pfister’s theorem (the existence problem of common zero points of polynomials of odd degree).

Structure of the set of Borel exceptional vectors for entire curves. II

We have obtained a description of structure of the sets of Picard and Borel exceptional vectors for transcendental entire curve in some sense. We consider only $p$-dimensional entire curves with linearly independent components without common zeros.In particular, the set of Borel exceptional vectors together with the zero vector is a finite union of subspaces in $\mathbb{C}^{p}$ of dimension at most $p-1$. Moreover, the sum of their dimensions does not exceed $p$ if anypairwise intersection of the subspaces contains only the zero vector. A similar result is also valid for the set of Picardexceptional vectors.Another result shows that the structure of the set of Borel exceptional vectors for an entire curve of integer orderdiffers somewhat from the structure of such a set for an entire curve of non-integer order.For a transcendental entire curve $\vec{G}:\mathbb{C}\to \mathbb{C}^{p}$ with linearly independent components and without common zeros having non-integer or zero order the set of Borel exceptional vectors together with the zero vector is a subspace in $\mathbb{C}^{p}$ of dimension at most $p-1$.
 However, the set of Picard exceptional vectors does not possess this property. We propose two examples of entire curves.The first example shows the set of Borel exceptional vectors together with the zero vector for $p$-dimensional entire curve of integer order isunion of subspaces of dimension at most $p-1$ such that the total sum of these dimensions does not exceed $p$ and intersection of any pair of these subspaces contains only zero vector. The set of Picard exceptional vectors for the curve has the same property.In the second example, we construct a $q$-dimensional entire curve of non-integer order for which the set of Borel exceptional vectors together with the zero vector is a subspace in $\mathbb{C}^{q}$ of dimension at most $q-1$ and the set of Picard exceptional vectors together with the zero vectordo not have the property. This set is a union of some subspaces.

Strong convergence theorem for common zero points of inverse strongly monotone mappings and common fixed points of generalized demimetric mappings

In this paper, we introduce a new and efficient algorithm for finding a common element of the set of common fixed points of a finite family of generalized demimetric mappings and the set of common zero points of a finite family of inverse strongly monotone mappings in Hilbert spaces. Strong convergence of the proposed algorithm is established under standard and mild conditions in a Hilbert space. As an application, we use our algorithm for solving the common solutions to the variational inequality problem, the common minimization problem, the multiple-set split feasibility problem, the split common fixed point problem and the split common null point problem.

Generalized Hamming weights of toric codes over hypersimplices and squarefree affine evaluation codes

Let $ \mathbb{F}_{q} $ be a finite field with $ q $ elements, where $ q $ is a power of a prime $ p $. A polynomial over $ \mathbb{F}_{q} $ is monomially squarefree if all its monomials are squarefree. In this paper, we determine an upper bound on the number of common zeroes of any set of $ r $ linearly independent monomially squarefree polynomials of $ \mathbb{F}_{q}[t_{1}, t_{2}, \dots, t_{s}] $ in the affine torus $ T = (\mathbb{F}_{q}^{*})^{s} $ under certain conditions on $ r $, $ s $ and the degree of these polynomials. Applying the results, we obtain the generalized Hamming weights of toric codes over hypersimplices and squarefree affine evaluation codes, as defined in [14].

An explicit solution to the weak Schottky problem

We give an explicit weak solution to the Schottky problem, in the spirit of Riemann and Schottky. For any genus $g$, we write down a collection of polynomials in genus $g$ theta constants, such that their common zero locus contains the locus of Jacobians of genus $g$ curves as an irreducible component. These polynomials arise by applying a specific Schottky-Jung proportionality to an explicit collection of quartic identities for theta constants in genus $g-1$, which are suitable linear combinations of Riemann's quartic relations.

A central quaternionic Nullstellensatz

Let I be a proper left ideal in the ring H[x1,…,xn] of polynomials in n central variables over the quaternion algebra H. Then there exists a point a=(a1,…,an)∈Hn with aiaj=ajai for all i,j, such that every polynomial in I vanishes at a. This generalizes a theorem of Jacobson, who proved the case n=1. Moreover, a polynomial f∈H[x1,…,xn] vanishes at all common zeroes of polynomials in I if and only if f belongs to the intersection of all completely prime left ideals that contain I – a notion introduced by Reyes in 2010.

Faber polynomials with common zero

We describe the set of meromorphic univalent functions in the class $$\Sigma $$ , for which the sequence of the Faber polynomials $$\{F_j\}_{j=1}^\infty $$ have the roots with following properties $$|F_n (z_0)|>0=\sum _{\begin{array}{c} j=1 j\not =n \end{array}}|F_j (z_0 )|$$ . For such functions we found an explicit form of the Faber polynomials as well as we discussed some properties.

Nearly irreducibility of polynomials and the Newton diagrams

Abstract Let f be a polynomial in two complex variables. We say that f is nearly irreducible if any two nonconstant polynomial factors of f have a common zero in C2. In the paper we give a criterion of nearly irreducibility for a given polynomial f in terms of its Newton diagram.

Linear Convergence of Split Equality Common Null Point Problem with Application to Optimization Problem

The purpose of this paper is to propose an iterative algorithm for solving the split equality common null point problem (SECNP), which is to find an element of the set of common zero points for a finite family of maximal monotone operators in Hilbert spaces. We introduce the concept of bounded linear regularity for the SECNP and construct several sufficient conditions to ensure the linear convergence of the algorithm. Moreover, some numerical experiments are given to test the validity of our results.

Strong convergence of a generalized forward–backward splitting method in reflexive Banach spaces

ABSTRACT In this paper, we study the so-called generalized monotone quasi-inclusion problem which is a generalization and extension of well-known monotone quasi-inclusion problem. We propose a forward–backward splitting method for solving this problem in the framework of reflexive Banach spaces. Based on Bregman distance function, we prove a strong convergence result of the proposed algorithm to a common zero of the problem. As an application, we apply the main result to the variational inequality problem. Finally, we provide some numerical examples to demonstrate our algorithm performance. The results presented in this paper improve and extend many known results in the literature.