Rota’s basis conjecture holds for random bases of vector spaces

Efficient Spectral-Chaos Methods for Uncertainty Quantification in Long-Time Response of Stochastic Dynamical Systems

A spherical t -design is a finite subset X in the unit sphere S n - 1 ⊂ R n which replaces the value of the integral on the sphere of any polynomial of degree at most t by the average of the values of the polynomial on the finite subset X . Generalizing the concept of spherical designs, Neumaier and Seidel (1988) defined the concept of Euclidean t -design in R n as a finite set X in R n for which ∑ i = 1 p ( w ( X i ) / ( | S i | ) ) ∫ S i f ( x ) d σ i ( x ) = ∑ x ∈ X w ( x ) f ( x ) holds for any polynomial f ( x ) of deg ( f ) ≤ t , where { S i , 1 ≤ i ≤ p } is the set of all the concentric spheres centered at the origin and intersect with X , X i = X ∩ S i , and w : X → R > 0 is a weight function of X . (The case of X ⊂ S n - 1 and with a constant weight corresponds to a spherical t -design.) Neumaier and Seidel (1988), Delsarte and Seidel (1989) proved the (Fisher type) lower bound for the cardinality of a Euclidean 2 e -design. Let Y be a subset of R n and let 𝒫 e ( Y ) be the vector space consisting of all the polynomials restricted to Y whose degrees are at most e . Then from the arguments given by Neumaier-Seidel and Delsarte-Seidel, it is easy to see that | X | ≥ dim ( 𝒫 e ( S ) ) holds, where S = ∪ i = 1 p S i . The actual lower bounds proved by Delsarte and Seidel are better than this in some special cases. However as designs on S , the bound dim ( 𝒫 e ( S ) ) is natural and universal. In this point of view, we call a Euclidean 2 e -design X with | X | = dim ( 𝒫 e ( S ) ) a tight 2 e -design on p concentric spheres. Moreover if dim ( 𝒫 e ( S ) ) = dim ( 𝒫 e ( R n ) ) ( = n + e e ) holds, then we call X a Euclidean tight 2 e -design. We study the properties of tight Euclidean 2 e -designs by applying the addition formula on the Euclidean space. Furthermore, we give the classification of Euclidean tight 4-designs with constant weight. It is possible to regard our main result as giving the classification of rotatable designs of degree 2 in R n in the sense of Box and Hunter (1957) with the possible minimum size n + 2 2 . We also give examples of nontrivial Euclidean tight 4-designs in R 2 with nonconstant weight,which give a counterexample to the conjecture of Neumaier and Seidel (1988) that there are no nontrivial Euclidean tight 2 e -designs even for the nonconstant weight case for 2 e ≥ 4 .

1. Algebraic geometry in KV. Let be an infinite field. K is given the so-called Zariski topology by defining a subset F to be closed if it is the set of common solutions of a collection of polynomial equations in n variables. Closed sets in K' are just the finite subsets. In K2, closed sets are finite unions of points and algebraic curves (e.g., the parabola y = x2 is an algebraic curve as it is the solutions of the polynomial equation y x2 = 0). The most unusual feature of this topology is that the open sets are all large. To be precise, each open set is dense, or equivalently, any two open sets have a non-empty intersection. This topology is not a T2-topology, although it is T1. In any topology, the subsets constructed from the open sets (or closed sets) by lattice operations (union, intersection, and complementation) are known as the constructible subsets. For the Zariski topology on , this Boolean algebra has a particularly nice interpretation, since if S is a constructible subset of Kn, then to decide the membership of a point x = (xI, ., xn) in S, one must simply check a finite number of polynomial equalities and inequalities involving the coordinates xi of x. For example, identify Mn (K) (n by n matrices over K) with Kn2 and let S be the set of all matrices having some fixed rank r. Let x = (xi1) and let mk,a(x) be a k th order minor determinant of x determined by the multi-index a. Then x belongs to S if and only if all mr+i,a(x) = 0 and some m,,,3(x) 4 0. Thus the somewhat unusual looking second condition of the theorem mentioned above is connected with the constructibility of a certain set of matrices. On K with the Zariski topology, the polynomial ring K[X,.. .,Xn] (denoted simply by K[X] most often) acts as a ring of continuous K-valued functions. This topological space, together with the ring of functions, is called affine n-space. More generally, a closed subset V in K is an affine variety if it is given the induced topology and a ring of continuous functions by restricting the polynomial functions on K to V. Each point of V has coordinates, and certain continuous functions are computed algebraically in terms of those coordinates. The ring of continuous functions is denoted by K[ V] and is called the coordinate ring of V. Certain open subsets of K are affine varieties by the following process. If f(X) belongs to K[X] then it defines a principal open subset n= {x in KI f(x) # 0}. K7n is homeomorphic to the closed set in n+ defined by the polynomial equation 1 Xn+1f(X1,...,Xn) = 0, under the mapping

The family of residue fields of a zero-dimensional commutative ring

A Generalization of an Addition Theorem of Kneser

Lie type algebras introduced in [1], [2] are natural generalizations of Lie algebras, associative algebras, Lie superalgebras with Z2-grading, and of algebras of some other classes. In [3], we studied a special case of algebras of Lie type called algebras of associative type. Recall the definition of an algebra of associative type. Let G be a semigroup, T a finite subset in G, A = ⊕ α∈T Aα a finite-dimensional G-graded algebra over a field k, i. e.,AαAβ ⊂ Aαβ if αβ ∈ T for α, β ∈ T andAαAβ = 0 if αβ / ∈ T . An algebraA is said to be an algebra of associative type if, for any α1, α2, α3 ∈ T , there exists λ = λ(α1, α2, α3) ∈ k, λ = 0, such that aα1(aα2aα3) = λ(aα1aα2)aα3 for any aαi ∈ Aαi , i = 1, 2, 3. Algebras generalizing semigroup algebras are examples of algebras of associative type. Let G be a finite semigroup. Denote by k[G] the vector space over k with basis eα, α ∈ G. We transform it into an algebra letting eαeβ = λα,βeαβ , λα,β ∈ k, λα,β = 0. Then the relation eα1(eα2eα3) = λ(eα1eα2)eα3 holds for λ = λα2,α3λα1,α2α3 λα1,α2λα1α2,α3 . In [3], an example of G-graded algebra B over Z was given such that, on the one hand, Q⊗

Matroids axiomatize the related notions of dimension and independence. We prove that if S is a set with k matroid structures, then S is the union of k subsets, the ith of which is independent in the ith matroid structure, iff for every (finite) subset A of S, $|A|$ is not larger than the sum of the dimensions of A in the k matroids. A matroid is representable if there is a dimension-preserving imbedding of it in a vector space. A matroid is constructed which is not the union of finitely many representable matroids. It is shown that a matroid is representable iff every finite subset of it is, and that if a matroid is representable over fields of characteristic p for infinitely many primes p, then it is representable over a field of characteristic 0. Similar results for other kinds of representation are obtained.

On Cullis’ determinant for rectangular matrices

The Greene–Magnanti theorem states that if is a finite matroid, and are bases and is a partition, then there is a partition such that is a base for every . The special case where each is a singleton can be rephrased as the existence of a perfect matching in the base transition graph. Pouzet conjectured that this remains true in infinite‐dimensional vector spaces. Later, he and Aharoni answered this conjecture affirmatively not just for vector spaces but also for infinite matroids. We prove two generalisations of their result. On the one hand, we show that ‘being a singleton’ can be relaxed to ‘being finite’ and this is sharp in the sense that the exclusion of infinite sets is really necessary. In addition, we prove that if and are bases, then there is a bijection between their finite subsets such that is a base for every . In contrast to the approach of Aharoni and Pouzet, our proofs are completely elementary, they do not rely on infinite matching theory.

On Isoperimetric Stability, Discrete Analysis 2018:14, 11 pp. Let $A$ be a subset of the Hamming cube $\{0,1\}^n$. If we regard the cube as a graph in the usual way, by joining two points $x$ and $y$ if they differ in exactly one coordinate, then the _edge boundary_ of $A$ is defined to be the set of all edges that join a point in $A$ to a point in its complement $\{0,1\}^n\setminus A$. The edge-isoperimetric inequality in the cube tells us that if $A$ has cardinality $2^d$, then the size of the edge boundary of $A$ is minimized when $A$ is a subcube of dimension $d$, in which case the number of edges between $A$ and its complement is $2^d(n-d)$. We can regard the graph as a Cayley graph, where the group is $\mathbb F_2^n$ and the generating set $S$ consists of the $n$ standard basis vectors. Then we find that the size of the edge boundary is always at least $|A|(|S|-\log_2|A|)$, an inequality that continues to hold when the size of $A$ is not a power of 2. As a consequence, if the size of the edge boundary is at most $(1-\gamma)|A||S|$, then $A$ must have size at least $2^{\gamma|S|}$. The purpose of this paper is to prove a similar statement for Abelian groups in general. Given an Abelian group $G$, a subset $A\subset G$, and a generating set $S$ of $G$, define the edge boundary to be the set of pairs $(a,s)\in A\times S$ such that $a+s\notin A$. The precise statement proved is that if the size of the edge boundary is at most $(1-\gamma)|A||S|$, then $A$ must have size at least $4^{(1-1/d)\gamma|S|}$, where $d$ is the exponent of $G$ -- that is, the smallest order of any element of $G$. Note that when $G=\mathbb F_2^n$ , then $d=2$, and we recover the statement above. Note also that in this case we can rewrite the lower bound for the size of $A$ as $|G|^\gamma$, a bound that is shown to be valid for groups of exponent 3 or 4 as well. However, for larger exponents the lower bound of $|G|^\gamma$ does not hold, and examples show that we have to settle for the weaker bound $4^{(1-1/d)\gamma|S|}$ proved in this paper. The reason that groups of low exponent are a special case is connected with the notion of a _dissociated set_, which is a subset $X$ of an Abelian group $G$ with the property that if $Y$ and $Z$ are finite subsets of $X$ with $\sum_{x\in Y}x=\sum_{x\in Z}x$, then $Y=Z$. Equivalently, it is a subset $X$ with a kind of linear independence property: if $x_1,\dots,x_n$ are distinct elements of $X$ and $\epsilon_1,\dots,\epsilon_n$ are coefficients taken from the set $\{-1,0,1\}$ such that $\sum_i\epsilon_ix_i=0$, then $\epsilon_1=\dots=\epsilon_n=0$. For groups of exponent 2 or 3, all possible integer coefficients can be replaced by ones that belong to the set $\{-1,0,1\}$, so this is just normal independence over the integers (and therefore, when these groups are considered as vector spaces, linear independence), but for groups of higher exponent that ceases to be the case. For example, in the group $\mathbb F_5^n$, the set $\{e_1,\dots,e_n,2e_1,\dots,2e_n\}$ is dissociated but not independent. These notions lead naturally to notions of dimension for subsets of Abelian groups: given a finite subset $E$, we define $\mathrm{dim}_D(E)$ to be the size of the largest dissociated subset of $E$ and $\mathrm{dim}_I(E)$ to be the size of the largest independent subset. A corollary of the main results of this paper is a sharp bound for the dimension of the _popular difference set_ of a set $A$, which is defined as follows. Given a parameter $\gamma>0$, we set $P_\gamma(A)$ to be the set of all group elements $g$ such that $|A\cap(A-g)|\geq\gamma|A|$. Popular difference sets play an important role in additive combinatorics. A result of Shkredov and Yekhanin states that $\mathrm{dim}_D(P_\gamma(A))$ is at most $C\gamma^{-1}\log|A|$, for some absolute constant $C$. (To see why the factor $\gamma^{-1}$ is necessary, consider an arbitrary union of $\gamma^{-1}$ subspaces of $\mathbb F_2^n$ of the same dimension.) In this paper it is shown that if $p$ is the smallest order of a non-zero element of $G$, then $\mathrm{dim}_I(P_\gamma(A)) \le (2(1-1/p))^{-1}\gamma^{-1}\log_2|A|$, which is refined to $\mathrm{dim}_I(P_\gamma(A)) \le \gamma^{-1}\log_3|A|$ in the special case $p=3$. These estimates are sharp when $G$ is homocyclic of exponent $p=2$ or $3$, when the two notions of dimension coincide. One of the proof ingredients in the paper is a result that stands on its own. A _down-set_ in $\mathbb Z_{\geq 0}^n$ is a subset $A$ with the property that if $x\in A$ and $y_i\leq x_i$ for every $i$, then $y\in A$. The paper gives an upper bound of $\frac 12\log_2|A|$ for the average number of non-zero coordinates of a vector in a down-set. It is a simple exercise to prove that this bound is sharp.

Matroids axiomatize the related notions of dimension and independence. We prove that if S is a set with k matroid structures, then S is the union of k subsets, the ith of which is independent in the ith matroid structure, iff for every (finite) subset A of S, $|A|$ is not larger than the sum of the dimensions of A in the k matroids. A matroid is representable if there is a dimension-preserving imbedding of it in a vector space. A matroid is constructed which is not the union of finitely many representable matroids. It is shown that a matroid is representable iff every finite subset of it is, and that if a matroid is representable over fields of characteristic p for infinitely many primes p, then it is representable over a field of characteristic 0. Similar results for other kinds of representation are obtained.

Higher reciprocity law and an analogue of the Grunwald–Wang theorem for the ring of polynomials over an ultra-finite field

On the independence numbers of a matroid

A geometric characterization of the structure of the group of automorphisms of an arbitrary Birkhoff-Grothendieck bundle splitting $\bigoplus_{i=1}^{r} \mathcal(m_{i})$ over $\mathbb{C}\mathbb{P}^{1}$ is provided, in terms of its action on a suitable space of generalized flags in the fibers over a finite subset $S\subset\mathbb{C}\mathbb{P}^{1}$. The relevance of such characterization derives from the possibility of constructing geometric models for diverse moduli spaces of stable objects in genus 0, such as parabolic bundles, parabolic Higgs bundles, and logarithmic connections, as collections of orbit spaces of parabolic structures and compatible geometric data satisfying a given stability criterion, under the actions of the different splitting types' automorphism groups, that are glued in a concrete fashion. We illustrate an instance of such idea, on the existence of several natural representatives for the induced actions on the corresponding vector spaces of (orbits of) logarithmic connections with residues adapted to a parabolic structure.

This paper is a study of certain geometric properties of convex sets in topological vector spaces (which are always assumed to be separated). These properties are closely related to fixed point theorems. While Theorems 7, 9, and 10 are explicitly fixed point or coincidence theorems for non-compact convex sets, Theorem 8 includes Theorem 9 (a coincidence theorem) as a special case. Theorems 1, 2, and 3 may be called "matching theorems", as the conclusions in these theorems assert the existence of a certain "matching". Consider for example, the conclusion of Theorem 3: "then there exists a non-empty finite subset {xl, x2 . . . . , x,} of X such that the convex hull of {x~, xz ..... x,} contains a point of the corresponding intersection (~ A(xi)". Notice that when n = 1, this becomes i=1 x~eA(x 0, a fixed point. The property required of the finite subset {Xl, x2 . . . . . x,} is a "matching" property involving two intuitively opposite conditions: a larger subset {xl, x2 . . . . . x,} of X would make its convex hull larger (and thus make it easier to have the required property), but would make the corresponding intersection