Kneser's Theorem Research Articles

On the solutions of the second-order $ (p, q) $-difference equation with an application to the fixed-point theory

<abstract><p>In this paper, we examined the existence and uniqueness of solutions to the second-order $ (p, q) $-difference equation with non-local boundary conditions by using the Banach fixed-point theorem. Moreover, we introduced a special case of this equation called the Euler-Cauchy-like $ (p, q) $-difference equation and provide its solution. We also studied the oscillation of solutions for this equation in $ (p, q) $-calculus and proved the $ (p, q) $-Sturm-type separation theorem and $ (p, q) $-Kneser theorem about the oscillation of solutions.</p></abstract>

Classifications and constructions of minimum size linear sets on the projective line

This paper aims to study linear sets of minimum size on the projective line, that is Fq-linear sets of rank k in PG(1,qn) admitting one point of weight one and having size qk−1+1. Examples of these linear sets have been found by Lunardon and the second author (2000) and, more recently, by Jena and Van de Voorde (2021). However, classification results for minimum size linear sets are known only for k≤5. In this paper we provide classification results for minimum size linear sets admitting two points with complementary weights. We construct new examples (not necessarily with complementary weights) and also study the related ΓL(2,qn)-equivalence issue. These results solve an open problem posed by Jena and Van de Voorde. The main tool relies on two results by Bachoc, Serra and Zémor (2017 and 2018) on the linear analogues of Kneser's and Vosper's theorems. We then conclude the paper pointing out a connection between critical pairs and linear sets, obtaining also some classification results for critical pairs.

Variations on the Tait–Kneser Theorem

At every point, a smooth plane curve can be approximated, to second order, by a circle; this circle is called osculating. One may think of the osculating circle as passing through three infinitesimally close points of the curve. A vertex of the curve is a point at which the osculating circle hyper-osculates: it approximates the curve to third order. Equivalently, a vertex is a critical point of the curvature function. Consider a (necessarily non-closed) curve, free from vertices. The classical Tait-Kneser theorem [13, 5] (see also [3, 10]), states that the osculating circles of the curve are pairwise disjoint, see Figure 1. This theorem is closely related to the four vertex theorem of S. Mukhopadhyaya [8] that a plane oval has at least 4 vertices (see again [3, 10]). Figure 1 illustrates the Tait-Kneser theorem: it shows an annulus foliated by osculating circles of a curve.

Product of primes in arithmetic progressions

We prove that, for all [Formula: see text] and for all invertible residue classes [Formula: see text] modulo [Formula: see text], there exists a natural number [Formula: see text] that is congruent to [Formula: see text] modulo [Formula: see text] and that is the product of exactly three primes, all of which are below [Formula: see text]. The proof is further supplemented with a self-contained proof of the special case of the Kneser Theorem we use.

Additive combinatorics methods in associative algebras

We adapt methods coming from additive combinatorics to the study of linear span in associative unital algebras. In particular, we establish for these algebras analogues of Kneser's theorem on sumsets and Tao's theorem on sets of small doubling. In passing we classify the finite-dimensional algebras over infinite fields with finitely many subalgebras. These algebras play a crucial role in our linear version of Kneser's theorem. We also explain how the original theorems for groups we linearize can be easily deduced from our results applied to group algebras without using Galois correspondence arguments.

Distance Sets on Circles

An n-point k-distance set on the unit sphere St ⊂ ℝt+1 is a set X of n points on S1 such that exactly k Euclidean distances occur between two distinct points in X. In this paper we treat distance sets on S1, and show that if k is sufficiently small relative to n, then X lies on a regular polygon. More precisely, we prove that for an n-point k-distance set X on S1 with n ≥ 4, if k > 3t or 3t — 2 according to whether n = 4t, 4t — 1 or n = 4t — 2, 4t — 3, respectively, then X lies on a regular 2k or (2k + 1)-sided polygon. Furthermore, we see that this bound cannot be further improved. In addition, we find an application of Kneser's addition theorem to distance sets on circles.

Colouring quadrangulations of projective spaces

A graph embedded in a surface with all faces of size 4 is known as a quadrangulation. We extend the definition of quadrangulation to higher dimensions, and prove that any graph G which embeds as a quadrangulation in the real projective space Pn has chromatic number n+2 or higher, unless G is bipartite. For n=2 this was proved by Youngs (1996) [20]. The family of quadrangulations of projective spaces includes all complete graphs, all Mycielski graphs, and certain graphs homomorphic to Schrijver graphs. As a corollary, we obtain a new proof of the Lovász–Kneser theorem.

Plünnecke and Kneser type theorems for dimension estimates

Given a division ring K containing the field k in its center and two finite subsets A and B of K*, we give some analogues of Plünnecke and Kneser Theorems for the dimension of the k-linear span of the Minkowski product AB in terms of the dimensions of the k-linear spans of A and B. We also explain how they imply the corresponding more classical theorems for abelian groups. These Plünnecke type estimates are then generalized to the case of associative algebras. We also obtain an analogue in the context of division rings of a theorem by Tao describing the sets of small doubling in a group.

Coloring hypercomplete and hyperpath graphs

Given a graph G with an induced subgraph H and a family F of graphs, we introduce a (hyper)graph HH(G;F)=(VH, EH), the hyper-H (hyper)graph of G with respect to F, whose vertices are induced copies of H in G, and \{H1,H2,\ldots,Hr\} \in EH if and only if the induced subgraph of G by the set \cupi=1r Hi is isomorphic to a graph F in the family F, and the integer r is the least integer for F with this property. When H is a k-complete or a k-path of G, we abbreviate HKk(G;F) and HPk(G;F) to Hk(G;F) and HPk(G;F), respectively. Our motivation to introduce this new (hyper)graph operator on graphs comes from the fact that the graph Hk(Kn;\{K2k\}) is isomorphic to the ordinary Kneser graph K(n;k) whenever 2k \leq n. As a generalization of the Lovasz--Kneser theorem, we prove that c(Hk(G;\{K2k\}))=c(G)-2k+2 for any graph G with w(G)=c(G) and any integer k\leq \lfloor w(G)/2\rfloor. We determine the clique and fractional chromatic numbers of Hk(G;\{K2k\}), and we consider the generalized Johnson graphs Hr(H;\{Kr+1\}) and show that c(Hr(H;\{Kr+1\}))\leq c(H) for any graph H and any integer r< w(H). By way of application, we construct examples of graphs such that the gap between their chromatic and fractional chromatic numbers is arbitrarily large. We further analyze the chromatic number of hyperpath (hyper)graphs HPk(G;Pm), and we provide upper bounds when m=k+1 and m=2k in terms of the k-distance chromatic number of the source graph.

Fourth-Order Four-Point Boundary Value Problem: A Solutions Funnel Approach

We investigate the existence of positive or a negative solution of several classes of four-point boundary-value problems for fourth-order ordinary differential equations. Although these problems do not always admit a (positive) Green's function, the obtained solution is still of definite sign. Furthermore, we prove the existence of an entire continuum of solutions. Our technique relies on the continuum property (connectedness and compactness) of the solutions funnel (Kneser's Theorem), combined with the corresponding vector field.

Strengthening of the Kneser theorem on zeros of solutions of the equation u″ + q(t)u = 0 using one functional equation

We present conditions under which a linear homogeneous second-order equation is nonoscillatory on a semiaxis and conditions under which its solutions have infinitely many zeros.

Invertible harmonic mappings beyond the Kneser theorem and quasiconformal harmonic mappings

Invertible harmonic mappings beyond the Kneser theorem and quasiconformal harmonic mappings

Characterizing the structure of [formula omitted] when [formula omitted] has small upper Banach density

Let A and B be two infinite sets of non-negative integers. Similar to Kneser's Theorem (Theorem 1.1 below) we characterize the structure of A + B when the upper Banach density of A + B is less than the sum of the upper Banach density of A and the upper Banach density of B.

ON A NON-COMPACT GENERALIZATION OF FAN'S MINIMAX THEOREM

In this paper, we first introduce the weak convexlike condition which generalizes the convexlike concept due to Fan. Next, using the separation theorem for convex sets, we will prove a non-compact generalization of Fan's minimax theorem by relaxing the concavelike assumption to the weak concavelike condition. Also we give some examples which show that the convex and concave assumptions on Kneser's minimax theorem can not be relaxed with the quasi-convex and quasi-concave conditions simultaneously, and the previous minimax theorems can not be available.

ON A NON-COMPACT GENERALIZATION OF FAN\'S MINIMAX THEOREM

In this paper, we first introduce the weak convexlike condition which generalizes the convexlike concept due to Fan. Next, using the separation theorem for convex sets, we will prove a non-compact generalization of Fan's minimax theorem by relaxing the concavelike assumption to the weak concavelike condition. Also we give some examples which show that the convex and concave assumptions on Kneser's minimax theorem can not be relaxed with the quasi-convex and quasi-concave conditions simultaneously, and the previous minimax theorems can not be available.

The Kneser property of the weak solutions of the three dimensional Navier-Stokes equations

The Kneser theorem for ordinary differential equations without uniqueness saysthat the attainability set is compact and connected at each instant of time.We establish corresponding results for the attainability set of weak solutionsfor the 3D Navier-Stokes equations satisfying an energy inequality. First, wepresent a simplified proof of our earlier result with respect to the weaktopology in the space $H$. Then we prove that this result also holds withrespect to the strong topology on $H$ provided that the weak solutionssatisfying the weak version of the energy inequality are continuous. Finally,using these results, we show the connectedness of the global attractor of afamily of setvalued semiflows generated by the weak solutions of the NSEsatisfying suitable properties.

THE ISOPERIMETRIC METHOD IN NON-ABELIAN GROUPS WITH AN APPLICATION TO OPTIMALLY SMALL SUMSETS

Set addition theory is born a few decades ago from additive number theory. Several difficult issues, more combinatorial in nature than algebraic, have been revealed. In particular, computing the values taken by the function: [Formula: see text] where G is a given group does not seem easy in general. Some successive results, using Kneser's Theorem, allowed the determination of the values of this function, provided that the group G is abelian. Recently, a method called isoperimetric, has been developed by Hamidoune and allowed new proofs and generalizations of the classical theorems in additive number theory. For instance, a new interpretation of the isoperimetric method has been able to give a new proof of Kneser's Theorem. The purpose of this article is to adapt this last proof in a non-abelian group, in order to give new values of the function μG, for some solvable groups and alternating groups. These values allow us in particular to answer negatively a question asked in the literature on the μG functions.

A generalization of Kneser's Addition Theorem

Let A = ( A 1 , … , A m ) be a sequence of finite subsets from an additive abelian group G. Let Σ ℓ ( A ) denote the set of all group elements representable as a sum of ℓ elements from distinct terms of A, and set H = stab ( Σ ℓ ( A ) ) = { g ∈ G : g + Σ ℓ ( A ) = Σ ℓ ( A ) } . Our main theorem is the following lower bound: | Σ ℓ ( A ) | ⩾ | H | ( 1 − ℓ + ∑ Q ∈ G / H min { ℓ , | { i ∈ { 1 , … , m } : A i ∩ Q ≠ ∅ } | } ) . In the special case when m = ℓ = 2 , this is equivalent to Kneser's Addition Theorem, and indeed we obtain a new proof of this result. The special case when every A i has size one is a new result concerning subsequence sums which extends some recent work of Bollobás–Leader, Hamidoune, Hamidoune–Ordaz–Ortuño, Grynkiewicz, and Gao, and resolves two recent conjectures of Gao, Thangadurai, and Zhuang.

Generalization of the Kneser theorem on zeros of solutions of the equation y″ + p(t)y = 0

We establish conditions for the oscillation of solutions of the equation y″ + p(t)Ay = 0 in a Banach space, where A is a bounded linear operator and p: ℝ+ → ℝ+ is a continuous function.

Conditions of oscillatory or nonoscillatory nature of solutions for a class of second-order semilinear differential equations

For a class of second-order semilinear differential equations, we prove the theorems on oscillatory or nonoscillatory nature of all proper solutions. These theorems are analogs of the well-known Kneser theorems for linear differential equations.