Integer-valued Functions Research Articles

Analysis of the two-for-one swap heuristic for approximating the maximum independent set in a k-polymatroid

Let f:2N→Z+ be a polymatroid (an integer-valued non-decreasing submodular set function with f(∅)=0). A k-polymatroid satisfies that f(e)≤k for all e∈N. We call S⊆Nindependent if f(S)=∑e∈Sf(e) and f(e)>0 for all e∈S. Such a set was also called a matching. Finding a maximum-size independent set in a 2-polymatroid has been studied and polynomial-time algorithms are known for linear polymatroids. For k≥3, the problem is NP-hard, and a ((2/k)−ϵ)-approximation is known and is obtained by swapping as long as possible a subset of up to (1/ϵ)logk−1⁡(2k+1) elements from the current solution by a set with one more element.Here we give a simple analysis of the more particular two-for-one repeated swapping heuristic, obtaining a tight (weaker) (2/(k+1))-approximation.

On rings of integer-valued rational functions

Let D ⊆ B be an extension of integral domains and E a subset of the quotient field of D. We introduce the ring of D -valued B-rational functions on E, denoted by Int B R ( E , D ) , which naturally extends the concept of integer-valued polynomials, defined as Int B R ( E , D ) : = { f ∈ B ( X ) ; f ( E ) ⊆ D } . The notion of Int B R ( E , D ) boils down to the usual notion of integer-valued rational functions when the subset E is infinite. In this paper, we aim to investigate various properties of these rings, such as prime ideals, localization, and the module structure. Furthermore, we study the transfer of some ring-theoretic properties from Int R ( E , D ) to D.

Factorization in rings of integer-valued rational functions

For a domain D, the ring Int ( D ) of integer-valued polynomials over D is atomic if D satisfies the ascending chain condition on principal ideals. However, even for a discrete valuation domain V, the ring Int R ( V ) of integer-valued rational functions over V is antimatter. We introduce a family of atomic rings of integer-valued rational functions and study various factorization properties in these rings.

Sufficient Conditions for a Graph k G Admitting all [ 1 , k ] -factors

Let g,f:V(G)→{0,1,2,3,⋯} be two functions satisfying g(x)≤f(x) for every x∈V(G). A (g,f)-factor of G is defined as a spanning subgraph F of G such that g(x)≤dF(x)≤f(x) for every x∈V(G). An (f,f)-factor is simply called an f-factor. Let φ be a nonnegative integer-valued function defined on V(G). Set Deveng,f={φ:g(x)≤φ(x)≤f(x) for every x∈V(G) and ∑x∈V(G)φ(x) is even}. If for each φ∈Deveng,f, G admits a φ-factor, then we say that G admits all (g,f)-factors. All (g,f)-factors are said to be all [1,k]-factors if g(x)≡1 and f(x)≡k for any x∈V(G). In this paper, we verify that for a connected multigraph G satisfying NG(X)=V(G) or |NG(X)|>(1+1k+1)|X|−1 for every X⊂V(G), kG admits all [1,k]-factors, where k≥2 is an integer and kG denotes the graph derived from G by replacing every edge of G with k parallel edges.

Skew-morphisms of elementary abelian 𝑝-groups

Abstract A skew-morphism of a finite group 𝐺 is a permutation 𝜎 on 𝐺 fixing the identity element, and for which there exists an integer-valued function 𝜋 on 𝐺 such that σ ⁢ ( x ⁢ y ) = σ ⁢ ( x ) ⁢ σ π ⁢ ( x ) ⁢ ( y ) \sigma(xy)=\sigma(x)\sigma^{\pi(x)}(y) for all x , y ∈ G x,y\in G . It is known that, for a given skew-morphism 𝜎 of 𝐺, the product of the left regular representation of 𝐺 with ⟨ σ ⟩ \langle\sigma\rangle forms a permutation group on 𝐺, called a skew-product group of 𝐺. In this paper, we study the skew-product groups 𝑋 of elementary abelian 𝑝-groups Z p n \mathbb{Z}_{p}^{n} . We prove that 𝑋 has a normal Sylow 𝑝-subgroup and determine the structure of 𝑋. In particular, we prove that Z p n ⊲ X \mathbb{Z}_{p}^{n}\lhd X if p = 2 p=2 and either Z p n ⊲ X \mathbb{Z}_{p}^{n}\lhd X or ( Z p n ) X ≅ Z p n − 1 (\mathbb{Z}_{p}^{n})_{X}\cong\mathbb{Z}_{p}^{n-1} if 𝑝 is an odd prime. As an application, for n ≤ 3 n\leq 3 , we prove that 𝑋 is isomorphic to a subgroup of the affine group AGL ⁢ ( n , p ) \mathrm{AGL}(n,p) and enumerate the number of skew-morphisms of Z p n \mathbb{Z}_{p}^{n} .

The Skolem property in rings of integer-valued rational functions

Let D be a domain and let Int(D) and IntR(D) be the ring of integer-valued polynomials and the ring of integer-valued rational functions, respectively. Skolem proved that if I is a finitely-generated ideal of Int(Z) with all the value ideals of I not being proper, then I=Int(Z). This is known as the Skolem property, which does not hold in Z[x]. One obstruction to Int(D) having the Skolem property is the existence of unit-valued polynomials. This is no longer an obstruction when we consider the Skolem property on IntR(D). We determine that the Skolem property on IntR(D) is equivalent to the maximal spectrum being contained in the ultrafilter closure of the set of maximal pointed ideals. We generalize the Skolem property using star operations and determine an analogous equivalence under this generalized notion.

Disorder-resistant fusion estimator design for nonlinear stochastic systems in the presence of measurement quantization

In this paper, we thoroughly study a new nonlinear disorder-resistant fusion estimation problem under both packet disorder and measurement quantization. Packet disorder, which occurs due mainly to random transmission delays (RTDs), are characterized via a series of integer-valued functions and random variables obeying certain distributions. Sensor measurements are logarithmically quantized before entering designated communication networks. The upper bounds on the filtering error covariances are acquired and later minimized by appropriately designing gain parameters of both local and fusion estimators. For fusion estimation, all local estimates and filtering covariances are integrated at a fusion center through the well-known federated criterion. Moreover, the fusion performance is examined by means of assessing the resistance of the designed estimator against the RTD-induced disorder. Simulation examples are presented to illustrate the applicability of the proposed disorder-resistant fusion estimator.

Height function localisation on trees

AbstractWe study two models of discrete height functions, that is, models of random integer-valued functions on the vertices of a tree. First, we consider the random homomorphism model, in which neighbours must have a height difference of exactly one. The local law is uniform by definition. We prove that the height variance of this model is bounded, uniformly over all boundary conditions (both in terms of location and boundary heights). This implies a strong notion of localisation, uniformly over all extremal Gibbs measures of the system. For the second model, we consider directed trees, in which each vertex has exactly one parent and at least two children. We consider the locally uniform law on height functions which are monotone, that is, such that the height of the parent vertex is always at least the height of the child vertex. We provide a complete classification of all extremal gradient Gibbs measures, and describe exactly the localisation-delocalisation transition for this model. Typical extremal gradient Gibbs measures are localised also in this case. Localisation in both models is consistent with the observation that the Gaussian free field is localised on trees, which is an immediate consequence of transience of the random walk.

Classification of Rank-One Submanifolds

We study ruled submanifolds of Euclidean space. First, to each (parametrized) ruled submanifold sigma , we associate an integer-valued function, called degree, measuring the extent to which sigma fails to be cylindrical. In particular, we show that if the degree is constant and equal to d, then the singularities of sigma can only occur along an (m-d)-dimensional “striction” submanifold. This result allows us to extend the standard classification of developable surfaces in {mathbb {R}}^{3} to the whole family of flat and ruled submanifolds without planar points, also known as rank-one: an open and dense subset of every rank-one submanifold is the union of cylindrical, conical, and tangent regions.

Generic length functions on countable groups

Let L(G) denote the space of integer-valued length functions on a countable group G endowed with the topology of pointwise convergence. Assuming that G does not satisfy any non-trivial mixed identity, we prove that a generic (in the Baire category sense) length function on G is a word length and the associated Cayley graph is isomorphic to a certain universal graph U independent of G. On the other hand, we show that every comeager subset of L(G) contains 2ℵ0 “asymptotically incomparable” length functions. A combination of these results yields 2ℵ0 pairwise non-equivalent regular representations G→Aut(U). We also prove that generic length functions are virtually indistinguishable from the model-theoretic point of view. Topological transitivity of the action of G on L(G) by conjugation plays a crucial role in the proof of the latter result.

Efficient Solution of Discrete Subproblems Arising in Integer Optimal Control with Total Variation Regularization

We consider a class of integer linear programs (IPs) that arise as discretizations of trust-region subproblems of a trust-region algorithm for the solution of control problems, where the control input is an integer-valued function on a one-dimensional domain and is regularized with a total variation term in the objective, which may be interpreted as a penalization of switching costs between different control modes. We prove that solving an instance of the considered problem class is equivalent to solving a resource-constrained shortest-path problem (RCSPP) on a layered directed acyclic graph. This structural finding yields an algorithmic solution approach based on topological sorting and corresponding run-time complexities that are quadratic in the number of discretization intervals of the underlying control problem, the main quantifier for the size of a problem instance. We also consider the solution of the RCSPP with an [Formula: see text] algorithm. Specifically, the analysis of a Lagrangian relaxation yields a consistent heuristic function for the [Formula: see text] algorithm and a preprocessing procedure, which can be employed to accelerate the [Formula: see text] algorithm for the RCSPP without losing optimality of the computed solution. We generate IP instances by executing the trust-region algorithm on several integer optimal control problems. The numerical results show that the accelerated [Formula: see text] algorithm and topological sorting outperform a general-purpose IP solver significantly. Moreover, the accelerated [Formula: see text] algorithm is able to outperform topological sorting for larger problem instances. We also give computational evidence that the performance of the superordinate trust-region algorithm may be improved if it is initialized with a solution obtained with the combinatorial integral approximation. History: Accepted by Andrea Lodi, Area Editor for Design and Analysis of Algorithms–Discrete. Supplemental Material: The online supplement is available at https://doi.org/10.1287/ijoc.2023.1294 .

A hyperbolic analogue of the Rademacher symbol

One of the most famous results of Dedekind is the transformation law of $$\log \Delta (z)$$ . After a half-century, Rademacher modified Dedekind’s result and introduced an $${{\,\textrm{SL}\,}}_2({\mathbb {Z}})$$ -conjugacy class invariant (integer-valued) function $$\Psi (\gamma )$$ called the Rademacher symbol. Inspired by Ghys’ work on modular knots, Duke–Imamoḡlu–Tóth (2017) constructed a hyperbolic analogue of the symbol. In this article, we study their hyperbolic analogue of the Rademacher symbol $$\Psi _\gamma (\sigma )$$ and provide its two types of explicit formulas by comparing it with the classical Rademacher symbol. In association with it, we contrastively show Kronecker limit type formulas of the parabolic, elliptic, and hyperbolic Eisenstein series. These limits give harmonic, polar harmonic, and locally harmonic Maass forms of weight 2.

About integer-valued variants of the theta and 6j symbols

This article contains essentially a rewriting of several properties of two well-known quantities, the so-called theta symbol (or triangular symbol), which is rational, and the 6 j symbol, which is usually irrational, in terms of two related integer-valued functions called gon and tet. Existence of these related integer-valued avatars, sharing most essential properties with their more popular partners, although a known fact, is often overlooked. The properties of gon and tet are easier to obtain, or to formulate, than those of the corresponding theta and 6 j symbols in both classical and quantum situations. Their evaluation is also simpler (this paper displays a number of explicit formulas and evaluation procedures that may speed up some computer programs). These two integer-valued functions are unusual in that their properties do not appear to be often discussed in the literature, but their features reflect those of related real-valued functions discussed in many places. Some of the properties that we shall discuss seem, however, to be new, in particular several relations between the function gon and inverse Hilbert matrices.

On the structure of the ring of integer-valued entire functions

The present paper studies the structure of the ring of integer-valued entire functions. We characterize certain classes of prime and maximal ideals and investigate some of their properties.

Sandpile Solitons in Higher Dimensions

Let $$p\in {\mathbb {Z}}^n$$ be a primitive vector and $$\Psi :{\mathbb {Z}}^n\rightarrow {\mathbb {Z}}, z\rightarrow \min (p\cdot z, 0)$$ . The theory of husking allows us to prove that there exists a pointwise minimal function among all integer-valued superharmonic functions equal to $$\Psi $$ “at infinity”. We apply this result to sandpile models on $${\mathbb {Z}}^n$$ . We prove existence of so-called solitons in a sandpile model, discovered in 2-dim setting by S. Caracciolo, G. Paoletti, and A. Sportiello and studied by the author and M. Shkolnikov in previous papers. We prove that, similarly to 2-dim case, sandpile states, defined using our husking procedure, move changeless when we apply the sandpile wave operator (that is why we call them solitons). We prove an analogous result for each lattice polytope A without lattice points except its vertices. Namely, for each function $$\begin{aligned} \Psi :{\mathbb {Z}}^n\rightarrow {\mathbb {Z}}, z\rightarrow \min _{p\in A\cap {\mathbb {Z}}^n}(p\cdot z+c_p), c_p\in {\mathbb {Z}}\end{aligned}$$ there exists a pointwise minimal function among all integer-valued superharmonic functions coinciding with $$\Psi $$ “at infinity”. The Laplacian of the latter function corresponds to what we observe when solitons, corresponding to the edges of A, intersect (see Fig. 1).

Eigenvalues and parity factors in graphs with given minimum degree

Let G be a graph and let g,f be nonnegative integer-valued functions defined on V(G) such that g(v)≤f(v) and g(v)≡f(v)(mod2) for all v∈V(G). A (g,f)-parity factor of G is a spanning subgraph H such that for each vertex v∈V(G), g(v)≤dH(v)≤f(v) and f(v)≡dH(v)(mod2). We prove sharp upper bounds for certain eigenvalues in an h-edge-connected graph G with given minimum degree to guarantee the existence of a (g,f)-parity factor.

Ideals of functions with compact support in the integer-valued case

Abstract For a zero-dimensional Hausdorff space X, denote, as usual, by C(X, ℤ) the ring of continuous integer-valued functions on X. If f ∈ C(X, ℤ), denote by Z(f) the set of all points of X that are mapped to 0 by f. The set $$\begin{array}{} \displaystyle C_K(X,\mathbb Z)=\{f\in C(X,\mathbb Z)\mid \text{cl}_X(X\smallsetminus \mathsf Z(f))\text{ is compact}\} \end{array}$$ is the integer-valued analogue of the ideal of functions with compact support in C(X). By first working in the category of locales and then interpreting the results in spaces, we characterize this ideal in several ways. Writing ζ X for the Banaschewski compactification of X, we also explore some properties of ideals of C(X, ℤ) associated with subspaces of ζ X analogously to how one associates, for any Tychonoff space Y, subsets of β Y with ideals of C(Y).

New invariants of Gromov–Hausdorff limits of Riemannian surfaces with curvature bounded below

Let \(\{X_i\}\) be a sequence of compact n-dimensional Alexandrov spaces (e.g. Riemannian manifolds) with curvature uniformly bounded below which converges in the Gromov–Hausdorff sense to a compact Alexandrov space X. The paper (Alesker in Arnold Math J 4(1):1–17, 2018) outlined (without a proof) a construction of an integer-valued function on X; this function carries additional geometric information on the sequence such as the limit of intrinsic volumes of the \(X_i\). In this paper we consider sequences of closed 2-surfaces and (1) prove the existence of such a function in this situation; and (2) classify the functions which may arise from the construction.

An Elementary Proof of Phase Transition in the Planar XY Model

Using elementary methods we obtain a power-law lower bound on the two-point function of the planar XY spin model at low temperatures. This was famously first rigorously obtained by Fröhlich and Spencer (Commun Math Phys 81(4):527–602, 1981) and establishes a Berezinskii–Kosterlitz–Thouless phase transition in the model. Our argument relies on a new loop representation of spin correlations, a recent result of Lammers (Probab Relat Fields, 2021) on delocalisation of general integer-valued height functions, and classical correlation inequalities.

How Many Directions Determine a Shape and other Sufficiency Results for Two Topological Transforms

In this paper we consider two topological transforms that are popular in applied topology: the Persistent Homology Transform and the Euler Characteristic Transform. Both of these transforms are of interest for their mathematical properties as well as their applications to science and engineering, because they provide a way of summarizing shapes in a topological, yet quantitative, way. Both transforms take a shape, viewed as a tame subset M M of R d \mathbb { R}^d , and associates to each direction v ∈ S d − 1 v\in S^{d-1} a shape summary obtained by scanning M M in the direction v v . These shape summaries are either persistence diagrams or piecewise constant integer-valued functions called Euler curves. By using an inversion theorem of Schapira, we show that both transforms are injective on the space of shapes, i.e. each shape has a unique transform. Moreover, we prove that these transforms determine continuous maps from the sphere to the space of persistence diagrams, equipped with any Wasserstein p p -distance, or the space of Euler curves, equipped with certain L p L^p norms. By making use of a stratified space structure on the sphere, induced by hyperplane divisions, we prove additional uniqueness results in terms of distributions on the space of Euler curves. Finally, our main result proves that any shape in a certain uncountable space of PL embedded shapes with plausible geometric bounds can be uniquely determined using only finitely many directions.