Integer Vectors Research Articles

Diagnosability of Vector Discrete-Event Systems Using Predicates

The diagnosability problem of faults is studied in the framework of vector discrete-event system (VDES). A VDES is a discrete-event system model in which a system state is represented by a vector with integer components, and state transitions are represented by integer vector addition. Predicates are employed to verify the fault diagnosability of VDES, since, defined as functions, predicates can conveniently identify particular state sets of interests. Specifically, system states are partitioned into different subsets by predicates, and the fault diagnosability of a system is verified by checking a subset of states. A sufficient condition for fault diagnosability of VDES is presented first. A necessary and sufficient condition is then developed. According to the two conditions, two types of predicates are given to partition the states in a VDES. In this work, a diagnoser or a full state enumeration is not constructed, whose complexity is exponential with respect to the system state size. In order to verify whether a system satisfies the proposed conditions, several polynomial algorithms and an algorithm by constructing a tree automaton are developed. Several examples are provided to illustrate the results obtained in this paper.

A consequence of the growth of rotation sets for families of diffeomorphisms of the torus

In this paper we consider $C^{\infty }$-generic families of area-preserving diffeomorphisms of the torus homotopic to the identity and their rotation sets. Let $f_{t}:\text{T}^{2}\rightarrow \text{T}^{2}$ be such a family, $\widetilde{f}_{t}:\mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ be a fixed family of lifts and $\unicode[STIX]{x1D70C}(\widetilde{f}_{t})$ be their rotation sets, which we assume to have interior for $t$ in a certain open interval $I$. We also assume that some rational point $(p/q,l/q)\in \unicode[STIX]{x2202}\unicode[STIX]{x1D70C}(\widetilde{f}_{\overline{t}})$ for a certain parameter $\overline{t}\in I$, and we want to understand the consequences of the following hypothesis: for all $t>\overline{t}$, $t\in I$, $(p/q,l/q)\in \text{int}(\unicode[STIX]{x1D70C}(\widetilde{f}_{t}))$. Under these very natural assumptions, we prove that there exists a $f_{\overline{t}}^{q}$-fixed hyperbolic saddle $P_{\overline{t}}$ such that its rotation vector is $(p/q,l/q)$. We also prove that there exists a sequence $t_{i}>\overline{t}$, $t_{i}\rightarrow \overline{t}$, such that if $P_{t}$ is the continuation of $P_{\overline{t}}$ with the parameter, then $W^{u}(\widetilde{P}_{t_{i}})$ (the unstable manifold) has quadratic tangencies with $W^{s}(\widetilde{P}_{t_{i}})+(c,d)$ (the stable manifold translated by $(c,d)$), where $\widetilde{P}_{t_{i}}$ is any lift of $P_{t_{i}}$ to the plane. In other words, $\widetilde{P}_{t_{i}}$ is a fixed point for $(\widetilde{f}_{t_{i}})^{q}-(p,l)$, and $(c,d)\neq (0,0)$ are certain integer vectors such that $W^{u}(\widetilde{P}_{\overline{t}})$ do not intersect $W^{s}(\widetilde{P}_{\overline{t}})+(c,d)$, and these tangencies become transverse as $t$ increases. We also prove that, for $t>\overline{t}$, $W^{u}(\widetilde{P}_{t})$ has transverse intersections with $W^{s}(\widetilde{P}_{t})+(a,b)$, for all integer vectors $(a,b)$, and thus one may consider that the tangencies above are associated to the birth of the heteroclinic intersections in the plane that do not exist for $t\leq \overline{t}$.

A Sequence Bounded Above by the Lucas Numbers

In this work, we consider the sequence whose nth term is the number of h-vectors of length n. The set of integer vectors E(n) is introduced. For, n >=2, the cardinality of E(n) is the nth Lucas number Ln is showed. The relation between the set of h-vectors L(n) and the set of integer vectors E(n) is given.

On the Recovery of an Integer Vector from Linear Measurements

Let 1 ≤ 2l ≤ m < d. A vector x ∈ ℤd is said to be l-sparse if it has at most l nonzero coordinates. Let an m × d matrix A be given. The problem of the recovery of an l-sparse vector x ∈ Zd from the vector y = Ax ∈ Rm is considered. In the case m = 2l, we obtain necessary conditions and sufficient conditions on the numbers m, d, and k ensuring the existence of an integer matrix A all of whose elements do not exceed k in absolute value which makes it possible to reconstruct l-sparse vectors in ℤd. For a fixed m, these conditions on d differ only by a logarithmic factor depending on k.

Simplices for numeral systems

The family of lattice simplices in R-n formed by the convex hull of the standard basis vectors together with a weakly decreasing vector of negative integers include simplices that play a central ro ...

Equidistribution of divergent orbits of the diagonal group in the space of lattices

We consider divergent orbits of the group of diagonal matrices in the space of lattices in Euclidean space. We define two natural numerical invariants of such orbits: the discriminant—an integer—and the type—an integer vector. We then study the question of the limit distributional behavior of these orbits as the discriminant goes to infinity. Using entropy methods we prove that, for divergent orbits of a specific type, virtually any sequence of orbits equidistributes as the discriminant goes to infinity. Using measure rigidity for higher-rank diagonal actions, we complement this result and show that, in dimension three or higher, only very few of these divergent orbits can spend all of their life-span in a given compact set before they diverge.

Counting arithmetical structures on paths and cycles

Let G be a finite, connected graph. An arithmetical structure on G is a pair of positive integer vectors d,r such that (diag(d)−A)r=0, where A is the adjacency matrix of G. We investigate the combinatorics of arithmetical structures on path and cycle graphs, as well as the associated critical groups (the torsion part of the cokernels of the matrices (diag(d)−A)). For paths, we prove that arithmetical structures are enumerated by the Catalan numbers, and we obtain refined enumeration results related to ballot sequences. For cycles, we prove that arithmetical structures are enumerated by the binomial coefficients 2n−1n−1, and we obtain refined enumeration results related to multisets. In addition, we determine the critical groups for all arithmetical structures on paths and cycles.

Arithmetical structures on graphs with connectivity one

Given a graph [Formula: see text], an arithmetical structure on [Formula: see text] is a pair of positive integer vectors [Formula: see text] such that [Formula: see text] and [Formula: see text] where [Formula: see text] is the adjacency matrix of [Formula: see text]. We describe the arithmetical structures on graph [Formula: see text] with a cut vertex [Formula: see text] in terms of the arithmetical structures on their blocks. More precisely, if [Formula: see text] are the induced subgraphs of [Formula: see text] obtained from each of the connected components of [Formula: see text] by adding the vertex [Formula: see text] and their incident edges, then the arithmetical structures on [Formula: see text] are in one to one correspondence with the [Formula: see text]-rational arithmetical structures on the [Formula: see text]’s. A rational arithmetical structure corresponds to an arithmetical structure where some of the integrality conditions are relaxed.

Reconfiguration of maximum-weight b-matchings in a graph

Consider a graph such that each vertex has a nonnegative integer capacity and each edge has a positive integer weight. Then, a b-matching in the graph is a multi-set of edges (represented by an integer vector on edges) such that the total number of edges incident to each vertex is at most the capacity of the vertex. In this paper, we study a reconfiguration variant for maximum-weight b-matchings: For two given maximum-weight b-matchings in a graph, we are asked to determine whether there exists a sequence of maximum-weight b-matchings in the graph between them, with subsequent b-matchings obtained by removing one edge and adding another. We show that this reconfiguration problem is solvable in polynomial time for instances with no integrality gap. Such instances include bipartite graphs with any capacity function on vertices, and 2-matchings in general graphs. Thus, our result implies that the reconfiguration problem for maximum-weight matchings can be solved in polynomial time for bipartite graphs.

Towers of generalized divisible quantum codes

A divisible binary classical code is one in which every code word has weight divisible by a fixed integer. If the divisor is $2^\nu$ for a positive integer $\nu$, then one can construct a Calderbank-Shor-Steane (CSS) code, where $X$-stabilizer space is the divisible classical code, that admits a transversal gate in the $\nu$-th level of Clifford hierarchy. We consider a generalization of the divisibility by allowing a coefficient vector of odd integers with which every code word has zero dot product modulo the divisor. In this generalized sense, we construct a CSS code with divisor $2^{\nu+1}$ and code distance $d$ from any CSS code of code distance $d$ and divisor $2^\nu$ where the transversal $X$ is a nontrivial logical operator. The encoding rate of the new code is approximately $d$ times smaller than that of the old code. In particular, for large $d$ and $\nu \ge 2$, our construction yields a CSS code of parameters $[[O(d^{\nu-1}), \Omega(d),d]]$ admitting a transversal gate at the $\nu$-th level of Clifford hierarchy. For our construction we introduce a conversion from magic state distillation protocols based on Clifford measurements to those based on codes with transversal $T$-gates. Our tower contains, as a subclass, generalized triply even CSS codes that have appeared in so-called gauge fixing or code switching methods.

On integer images of max-plus linear mappings

Let us extend the pair of operations ⊕,⊗=max,+ over real numbers to matrices in the same way as in conventional linear algebra.We study integer images of mappings x→A⊗x, where A∈Rm×n and x∈Rn. The question whether A⊗x is an integer vector for at least one x∈Rn has been studied for some time but polynomial solution methods seem to exist only in special cases. In the terminology of combinatorial matrix theory this question reads: is it possible to add constants to the columns of a given matrix so that all row maxima are integer? This problem has been motivated by attempts to solve a class of job-scheduling problems.We present two polynomially solvable special cases aiming to move closer to a polynomial solution method in the general case.

Subset Vector Perturbation Pre-Coding for MU-MIMO Downlink Systems

Multi-user multiple input multiple output (MU-MIMO) is a basic technique which has been widely investigated for its ability to increase system capacity. However, the inter-user interference (IUI) is one of the obstacles to obtain system capacity. With the help of pre-coding technique, the IUI can be canceled or reduced at the base station (BS). Vector perturbation (VP) is one of the most celebrated pre-coding schemes that can approach system capacity. By perturbing users’ data vector with a certain integer vector offset at the BS, the VP scheme can offer the users a higher effective signal-to-noise ratio. To reach the theory bound of VP, the BS has to select the optimal integer vector from all candidate integer vectors globally. This exhaustive search has been proved to be NP-hard, which is hard to be implemented. In this paper, we analyze several state-of-the-art modified low-complexity VP schemes and propose the subset VP (SVP) scheme to reduce the complexity of the VP. The basic principle of the SVP is that some vectors are selected as the optimal vector with high probability, whereas the others are scarcely selected. So, we try to form a subset consists of vectors with a relatively higher probability. First, we calculate the probability of all candidate vectors which have been selected as the optimal vector. Second, we propose a method to divide all vectors into several subgroups. Vectors in the same subgroup have a similar probability, whereas vectors in different subgroups have distinguished probability. Third, we build a subset that is comprised of several subgroups in which vectors have a relatively higher probability. Simulation results show that we get almost the same bit-error rate performance with a significant complexity reduction.

On the Recovery of an Integer Vector from Linear Measurements

Пусть $1\le 2l\le m&lt;d$. Мы говорим, что вектор $x\in\mathbb Z^d$ является $l$-разреженным, если он имеет не более $l$ ненулевых координат. Пусть задана $(m\times d)$-матрица $A$. Рассматривается задача восстановления $l$-разреженного вектора $x\in\mathbb Z^d$ по вектору $y=A x\in\mathbb R^m$. В случае $m=2l$ мы находим необходимые условия и достаточные условия на числа $m$, $d$, $k$ для того, чтобы существовала целочисленная матрица $A$, все элементы которой по модулю не превосходят $k$, позволяющая восстановить $l$-разреженные векторы в $\mathbb Z^d$. При фиксированном $m$ эти условия на $d$ отличаются лишь логарифмическим множителем по $k$. Библиография: 6 названий.

Design of lattice‐based ElGamal encryption and signature schemes using SIS problem

AbstractThe emerging technologies like quantum computing and cloud computing have a complex computing power and can break the security of the classical cryptographic constructions (like ElGamal's cryptosystem). Thus, we were motivated to work on hard lattice problems, which are more complex and can resist these new modern technologies. In this proposal, we present a lattice‐based version of discrete logarithm problem‐based ElGamal public‐key encryption and signature schemes that exhibit strong security features and efficient implementation. The security of these presented schemes is based on the worst‐case hardness of approximating a small integer vector in their corresponding lattice. Furthermore, this proposed work illustrates a security proof of the proposed schemes and shows that the presented schemes are well protected in the modern computing environment. The performance analysis and comparison with classical ElGamal schemes and recent lattice‐based schemes are also made, and it is seen that the proposed schemes achieve better performance.

Synthesis of generalized neural elements by means of the tolerance matrices

Application of neuromorphic structures in various spheres of human activity on the basis of generalized neural elements will become possible if effective methods for verifying realizability of the logic algebra functions by one neuron element with a generalized threshold activation function and synthesis of such elements with a large number of edntries are developed. A notion of nucleus of Boolean functions in relation to a given system of characters was introduced and algebraic structure of nuclei and reduced nuclei of Boolean functions was investigated. Relation between the nuclei of the logic algebra functions which are realized by one generalized neural element and matrices of tolerance was established. It was shown that the Boolean function is realized by one generalized neuron element if and only if the nucleus of this function admits representation by the matrices of tolerance. If there is no nucleus relative to a specified system of characters for a Boolean function, then such a function is not realized by one generalized neural element in relation to a specified system of characters. On the basis of the properties of the matrices of tolerance, a number of necessary and sufficient conditions for realization of the logic algebra functions by one generalized neural element were obtained. Based on the sufficient conditions, an algorithm for synthesis of integer-valued generalized neural elements with a large number of entries was constructed. In the synthesis of integer-valued generic neural elements for realization of the logic algebra functions, a block representation of the Boolean function nucleus was used and based on the properties of the matrices of tolerance, coordinates of the integer vector of the structure of the generalized neural element were sequentially found

A polynomial time algorithm for a variant of shortest vector problem with applications in compute-and-forward design

In wireless communication networks, one of the most significant problems in Compute-and-Forward (CPF) strategy based on Integer-Forcing (IF) technique is to search the optimal integer coefficient vector for a relay to decode the received messages. While this is a variant of shortest vector problem (SVP), which is prohibitively complex in its general form. In this paper, we mainly focus on the optimal integer solution with non-zero entries in multi-source one-relay model. For comparison we extend the number (or dimension) of communication nodes from 2 to any larger integer n based on an existing 2-dimensional search algorithm with non-zero entries. Then we add this non-zero-entries constraint to an existing search algorithm and make some improvements to search the optimal integer vector by solving an optimization problem over only one variable instead of searching all vector components, which has a much lower time complexity. We prove that our new search algorithm is more efficient with polynomial complexity compared with the extended n-dimensional algorithm, meanwhile the same optimal integer solution is guaranteed.

Topology of a class of [formula omitted]-crystallographicreplication tiles

We study the topological properties of a class of planar crystallographic replication tiles. Let M∈Z2×2 be an expanding matrix with characteristic polynomial x2+Ax+B (A,B∈Z, B≥2) and v∈Z2 such that (v,Mv) are linearly independent. Then the equation MT+B−12v=T∪(T+v)∪(T+2v)∪⋯∪(T+(B−2)v)∪(−T)defines a unique nonempty compact set T satisfying To¯=T. Moreover, T tiles the plane by the crystallographic group p2 generated by the π-rotation and the translations by integer vectors. It was proved by Leung and Lau in the context of self-affine lattice tiles with collinear digit set that T∪(−T) is homeomorphic to a closed disk if and only if 2|A|<B+3. However, this characterization does not hold anymore for T itself. In this paper, we completely characterize the tiles T of this class that are homeomorphic to a closed disk.

Largest Minimal Inversion-Complete and Pair-Complete Sets of Permutations

We solve two related extremal problems in the theory of permutations. A set 𝑄 of permutations of the integers 1 to 𝑛 is inversion-complete (resp., pair-complete) if for every inversion (𝑗, 𝑖), where 1 ≤ 𝑖 < 𝑗 ≤ 𝑛, (resp., for every pair (𝑖, 𝑗), where 𝑖 ≠ 𝑗) there exists a permutation in 𝑄 where 𝑗 is before 𝑖. It is minimally inversion-complete if in addition no proper subset of 𝑄 is inversion-complete; and similarly for pair completeness. The problems we consider are to determine the maximum cardinality of a minimal inversion-complete set of permutations, and that of a minimal pair-complete set of permutations. The latter problem arises in the determination of the Caratheodory numbers for certain abstract convexity structures on the (𝑛 − 1)-dimensional real and integer vector spaces. Using Mantel's Theorem on the maximum number of edges in a triangle-free graph, we determine these two maximum cardinalities and we present a complete description of the optimal sets of permutations for each problem. Perhaps surprisingly (since there are twice as many pairs to cover as inversions), these two maximum cardinalities coincide whenever 𝑛 ≥ 4.

Calculation of Gaussian Quadrature with High Accuracy Mathematics Package

We are checking here the dependence of numerical integration accuracy on the quantity of integration points and the accuracy of machine representation of numbers. For this purpose, the package HAHMath is applied. This package allows one to carry out calculations with arbitrary long machine decimal numbers, presented as vectors of integers. Integrals are substituted for Gaussian sums where Hermite polynomials zeros and corresponding weights, computed by the same package are used. It is shown that the chosen case accuracy of final calculations depends on the used machine numbers’ length more strictly than on the quantity of the integration points.

Success Probability of the Babai Estimators for Box-Constrained Integer Linear Models

In many applications including communications, one may encounter a linear model where the parameter vector $\hbx$ is an integer vector in a box. To estimate $\hbx$, a typical method is to solve a box-constrained integer least squares (BILS) problem. However, due to its high complexity, the box-constrained Babai integer point $\x^\sBB$ is commonly used as a suboptimal solution. In this paper, we first derive formulas for the success probability $P^\sBB$ of $\x^\sBB$ and the success probability $P^\sOB$ of the ordinary Babai integer point $\x^\sOB$ when $\hbx$ is uniformly distributed over the constraint box. Some properties of $P^\sBB$ and $P^\sOB$ and the relationship between them are studied. Then, we investigate the effects of some column permutation strategies on $\P^\sBB$. In addition to V-BLAST and SQRD, we also consider the permutation strategy involved in the LLL lattice reduction, to be referred to as LLL-P. On the one hand, we show that when the noise is relatively small, LLL-P always increases $P^\sBB$ and argue why both V-BLAST and SQRD often increase $P^\sBB$; and on the other hand, we show that when the noise is relatively large, LLL-P always decreases $P^\sBB$ and argue why both V-BLAST and SQRD often decrease $P^\sBB$. We also derive a column permutation invariant bound on $P^\sBB$, which is an upper bound and a lower bound under these two opposite conditions, respectively. Numerical results demonstrate our findings. Finally, we consider a conjecture concerning $\x^\sOB$ proposed by Ma et al. We first construct an example to show that the conjecture does not hold in general, and then show that it does hold under some conditions.