Invariant measures for non-primitive tiling substitutions

We consider self-affine tiling substitutions in Euclidean space and the corresponding tiling dynamical systems. It is well known that in the primitive case, the dynamical system is uniquely ergodic. We investigate invariant measures when the substitution is not primitive and the tiling dynamical system is non-minimal. We prove that all ergodic invariant probability measures are supported on minimal components, but there are other natural ergodic invariant measures, which are infinite. Under some mild assumptions, we completely characterize σ-finite invariant measures which are positive and finite on a cylinder set. A key step is to establish recognizability of non-periodic tilings in our setting. Examples include the “integer Sierpiński gasket and carpet” tilings. For such tilings, the only invariant probability measure is supported on trivial periodic tilings, but there is a fully supported σ-finite invariant measure that is locally finite and unique up to scaling.     

Komlós's tiling theorem via graphon covers

Komlós [Komlós: Tiling Turán Theorems, Combinatorica, 2000] determined the asymptotically optimal minimum-degree condition for covering a given proportion of vertices of a host graph by vertex-disjoint copies of a fixed graph H, thus essentially extending the Hajnal–Szemerédi theorem that deals with the case when H is a clique. We give a proof of a graphon version of Komlós's theorem. To prove this graphon version, and also to deduce from it the original statement about finite graphs, we use the machinery introduced in [Hladký, Hu, Piguet: Tilings in graphons, arXiv:1606.03113]. We further prove a stability version of Komlós's theorem.     

Nonfinite basicity for a certain number system

A negative answer is given to the question whether there exists a finite basis for real numbers with multiplication and a unary operation 1 − x.     

Factoring groups and tiling space

The problem of tiling space by translates of certain star bodies, called crosses and semicrosses, is intimately connected with finding a subsetA of a finite abelian groupG such that for a particular subset of the integersS each non-zero element ofG is uniquely expressible in the forms·g withs inS andg inA. This paper examines some of the algebraic questions raised; in particular it obtains bounds on the number of elements inS, constructs factorizations ofZ _p ⁿ , and presents an example of a setS that factors no group.     

Some matrix equations over a finite field

Summary Explicit formulas are found for the number of solutions over a finite field of several matrix equations, for example:X′A+A′X=B. Conditons for solvability are also given. The research for thispaper was supported by National Science Foundation Research Grant G-2990.     

Comparing Semigroup and Monoid Presentations for Finite Monoids

 It is known that for any finite group G given by a finite group presentation there exists a finite semigroup presentation for G of the same deficiency, i.e. satisfying . It is also known that the analogous statement does not hold for all finite monoids. In this paper we give a necessary and sufficient condition for a finite monoid M, given by a finite monoid presentation, to have a finite semigroup presentation of the same deficiency.     

Homological identities for a finite homogeneous graph

An application of the Atiyah-Bott trace identity to the study of the spectral characteristics of a finite (q+1)-homogeneous factorgraph Y=Γ/X is given (X is an infinite (q+1)-homogeneous tree, Γ a free group of isometries of X). Bibliography: 9 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 205, 1993, pp. 110–121.     

Persistence of a two-dimensional super-Brownian motion in a catalytic medium

Summary. The super-Brownian motion X ^ϱ in a super-Brownian medium ϱ constructed in [DF97a] is known to be persistent (no loss of expected mass in the longtime behaviour) in dimensions one ([DF97a]) and three ([DF97b]). Here we fill the gap in showing that persistence holds also in the critical dimension two. The key to this result is that in any dimension (d≤3), given the catalyst, the variance of the process is finite `uniformly in time'. This is in contrast to the `classical' super-Brownian motion where this holds only in high dimensions (d≥3), whereas in low dimensions the variances grow without bound, and the process clusters leading to local extinction. Received: 21 November 1996 / In revised form: 31 March 1997     

Comparing Semigroup and Monoid Presentations for Finite Monoids

 It is known that for any finite group G given by a finite group presentation there exists a finite semigroup presentation for G of the same deficiency, i.e. satisfying . It is also known that the analogous statement does not hold for all finite monoids. In this paper we give a necessary and sufficient condition for a finite monoid M, given by a finite monoid presentation, to have a finite semigroup presentation of the same deficiency. (Received 17 April 2001; in revised form 15 September 2001)     

On collineations and dualities of finite generalized polygons

In this paper we generalize a result of Benson to all finite generalized polygons. In particular, given a collineation θ of a finite generalized polygon S, we obtain a relation between the parameters of S and, for various natural numbers i, the number of points x which are mapped to a point at distance i from x by θ. As a special case we consider generalized 2n-gons of order (1, t) and determine, in the generic case, the exact number of absolute points of a given duality of the underlying generalized n-gon of order t.     

On a class of nonlocal parabolic equations

We consider a boundary value problem for parabolic equations with nonlocal nonlinearity of such a form that favorably differs from other equations in that it leads to partial differential equations that have important properties of ordinary differential equations. Local solvability and uniqueness theorems are proved, and an analog of the Painlevé singular nonfixed points theorem is proved. In this case, there is an alternative—either a solution exists for all t ≥ 0 or it goes to infinity in a finite time t = T (blowup mode). Sufficient conditions for the existence of a blowup mode are given.     

Proof of a tiling conjecture of Komlós

A conjecture of Komlós states that for every graph H, there is a constant K such that if G is any n‐vertex graph of minimum degree at least (1 ? (1/χ_cr(H)))n, where χ_cr(H) denotes the critical chromatic number of H, then G contains an H‐matching that covers all but at most K vertices of G. In this paper we prove that the conjecture holds for all sufficiently large values of n. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 23: 180–205, 2003     

Diameter and diametrical pairs of points in ultrametric spaces

Let F(X) be the set of finite nonempty subsets of a set X. We have found the necessary and sufficient conditions under which for a given function τ: F(X) → ℝ there is an ultrametric on X such that τ(A) = diamA for every A ∈ F(X). For finite nondegenerate ultrametric spaces (X, d) it is shown that X together with the subset of diametrical pairs of points of X forms a complete k-partite graph, k ⩾ 2, and, conversely, every finite complete k-partite graph with k ⩾ 2 can be obtained by this way. We use this result to characterize the finite ultrametric spaces (X, d) having the minimal card{(x, y): d(x, y) = diamX, x, y ∈ X} for given card X.     

Singular points of a star-finite tiling

Let be a star-finite tiling of a topological vector space of dimension greater than one and let S() denote the set of singular points of . We show that S() is either uncountable or empty by investigating the density in S() of certain subcollections of geometrically interesting singular points.     

The generic fiber of the universal deformation space associated to a tame Galois representation

We will study the generic fiber over of the universal deformation ring R _Q , as defined by Mazur, for deformations unramified outside a finite set of primes Q of a given Galois representation , E a number field, k a finite field of characteristic l. The main result will be that, if ˉρ is tame and absolutely irreducible, and if one assumes the Leopoldt conjecture for the splitting field E ₀ of , then defines a smooth l-adic analytic variety, near the trivial lift ρ₀ of ˉρ, whose dimension is given by cohomological constraints and as predicted by Mazur. As a corollary it follows that, in the cases considered here, R _Q is a quotient of by an ideal I generated by exactly m equations, where and . Under the above assumptions for and ˉρ odd, using ideas of Coleman, Gouvêa and Mazur it should now be possible to show that modular points are Zariski-dense in the component of , that contains the trivial lift ρ₀, provided this lift satisfies the Artin conjecture and E ₀ satisfies the Leopoldt conjecture. Furthermore, in the Borel case, we show that the Krull dimension of R _Q can exceed any given number, provided Q is chosen appropriately. At the same time, we present some evidence that despite this fact, one might however expect that the dimension of the generic fiber is given by the same cohomological formula as in the tame case. Received: 12 December 1997 / Revised version: 5 February 1998     

Optimal boundary control of vibrations at one endpoint of a string when the second endpoint is free

The generalized solution u(x, t) of the wave equation u _tt (x, t) − u _xx (x, t) = 0 admitting the existence of finite energy at every time instant t is used to find among all W ₂ ¹ [0,T]-functions with a long time interval T the optimal boundary control for a string with a free endpoint that takes the vibration process from a given arbitrary state to a given final state. __________ Translated from Nelineinaya Dinamika i Upravlenie, No. 4, pp. 23–36, 2004.     

On a problem involving bounded epimorphisms of lattices

A latticeL satisfies thebounded epimorphism condition if wheneverM is a lattce and ϕ:M →L is a bounded epimorphism, there exists a homomorphismι:L →M such that ιϕ=id _L . we show that the class of finite lattices satisfying the bounded epimorphism condition is properly contained in the class of finite lattices satisfying Whitman's condition (W). We also introduce a property defined for finite lattices that is sufficient to imply the bounded epimorphism condition. Presented by B. Jónsson.     

A characterization of vector measure games inpNA

We give a complete characterization of games inpNA of the formf ^o μ (where μ is a vector of finite number of non-atomic probability measures, andf is a real valued function on the range of μ withf(0)=0). Specifically, we show thatf ^o μ is inpNA iff “f is continuous at μ” (the definition of the latter is given in the paper). This work was supported by National Science Foundation Grant SOC 75-21820-A01 at the Institute for Mathematical Studies in the Social Sciences, Stanford University.