Simplicial complexes with rigid depth

We extend a result of Minh and Trung (Adv. Math. 226:1285–1306, 2011) to get criteria for depth ${I = \rm {depth}\sqrt{I}}$ , where I is an unmixed monomial ideal of the polynomial ring S?=?K[x ₁, . . . , x _n ]. As an application we characterize all the pure simplicial complexes Δ which have rigid depth, that is, which satisfy the condition that for every unmixed monomial ideal ${I\subset S}$ with ${\sqrt{I}=I_\Delta}$ one has depth(I)?=?depth(I _Δ).     

Higher complexity search problems for bounded arithmetic and a formalized no-gap theorem

We give a new characterization of the strict $\forall {\Sigma^b_j}$ sentences provable using ${\Sigma^b_k}$ induction, for 1 ?? j ?? k. As a small application we show that, in a certain sense, Buss??s witnessing theorem for strict ${\Sigma^b_k}$ formulas already holds over the relatively weak theory PV. We exhibit a combinatorial principle with the property that a lower bound for it in constant-depth Frege would imply that the narrow CNFs with short depth j Frege refutations form a strict hierarchy with j, and hence that the relativized bounded arithmetic hierarchy can be separated by a family of $\forall {\Sigma^b_1}$ sentences.     

The universality theorem for Hecke L-functions

We extend the universality theorem for Hecke L-functions attached to ray class characters from the previously known strip ${ \max \{\frac{1}{2}, 1-\frac{1}{d}\} < {\rm Re}\,s < 1}$ for ${d=\left[K:\mathbb{Q}\right]}$ to the maximal strip ${\frac{1}{2} < {\rm Re}\,s < 1}$ under an assumption of a weak version of the density hypothesis. As a corollary, we give a new proof of the universality theorem for the Dedekind zeta function ζ _K (s) in the case of ${K/\mathbb{Q}}$ finite abelian.     

A law of the iterated logarithm for geometrically weighted martingale difference sequences

We consider random variables ξ(β)=Σ _n=0 ^∞ β ⁿ Y _n for β<1. We prove that if the(Y _n)_n∈N is a stationary ergodic martingale difference sequence andE Y ₀ ² =1, then the following law of the iterated logarithm holds: $$\mathop {\lim \sup }\limits_{\beta \nearrow 1} \frac{{\sqrt {1 - \beta ^2 } }}{{\sqrt {2\log \log \frac{1}{{1 - \beta ^2 }}} }}\xi (\beta ) = 1 a.s.$$ We prove also the corresponding Central Limit Theorem. This generalizes a theorem by Bovier and Picco where the i.i.d. case was studied.     

On Weak*-Convergence in \boldsymbol{H^1_L(\boldsymbol{\mathbb R}^d)}

Let L?=???Δ?+?V be a Schrödinger operator on $\mathbb R^d$ , d?≥?3, where V is a nonnegative function, $V\ne 0$ , and belongs to the reverse Hölder class RH _d/2. In this paper, we prove a version of the classical theorem of Jones and Journé on weak*-convergence in the Hardy space $H^1_L(\mathbb R^d)$ .     

Derived Equivalences of Actions of a Category

Let $\mathbb{k}$ be a commutative ring and I a category. As a generalization of a $\mathbb{k}$ -category with a (pseudo) action of a group we consider a family of $\mathbb{k}$ -categories with a (pseudo, lax, or oplax) action of I, namely an oplax functor from I to the 2-category of small $\mathbb{k}$ -categories. We investigate derived equivalences of those oplax functors, and establish a Morita type theorem for them. This gives a base of investigations of derived equivalences of Grothendieck constructions of those oplax functors.     

Stanley depth of weakly polymatroidal ideals

Let \({\mathbb{K}}\) be a field and \({S = \mathbb{K}[x_1,\ldots,x_n]}\) be the polynomial ring in n variables over the field \({\mathbb{K}}\) . In this paper, it is shown that Stanley’s conjecture holds for S/I if I is a weakly polymatroidal ideal.     

L-spaces and the P-ideal dichotomy

We extend a theorem of Todor?evi?: Under the assumption ( $ \mathcal{K} $ ) (see Definition 1.11), $$ \boxtimes \left\{ \begin{gathered} any regular space Z with countable tightness such that \hfill \\ Z^n is Lindel\ddot of for all n \in \omega has no L - subspace. \hfill \\ \end{gathered} \right. $$ We assume $ \mathfrak{p} $ > ω ₁ and a weak form of Abraham and Todor?evi?’s P-ideal dichotomy instead and get the same conclusion. Then we show that $ \mathfrak{p} $ > ω ₁ and the dichotomy principle for P-ideals that have at most ?₁ generators together with ? do not imply that every Aronszajn tree is special, and hence do not imply (ie1-4). So we really extended the mentioned theorem.     

On the Stanley depth of weakly polymatroidal ideals

Let ${\mathbb{K}}$ be a field and ${S = \mathbb{K}[x_1,\dots,x_n]}$ be the polynomial ring in n variables over the field ${\mathbb{K}}$ . In this paper, it is shown that Stanley’s conjecture holds for I and S/I if I is a product of monomial prime ideals or I is a high enough power of a polymatroidal or a stable ideal generated in a single degree.     

On the Equality Problem of Conjugate Means

Let ${I\subset\mathbb{R}}$ be a nonvoid open interval and let L : I ²→ I be a fixed strict mean. A function M : I ²→ I is said to be an L-conjugate mean on I if there exist ${p,q\in\,]0,1]}$ and ${\varphi\in CM(I)}$ such that $$M(x,y):=\varphi^{-1}(p\varphi(x)+q\varphi(y)+(1-p-q) \varphi(L(x,y)))=:L_\varphi^{(p,q)}(x,y),$$ for all ${x,y\in I}$ . Here L(x, y) : = A _χ(x, y) ${(x,y\in I)}$ is a fixed quasi-arithmetic mean with the fixed generating function ${\chi\in CM(I)}$ . We examine the following question: which L-conjugate means are weighted quasi-arithmetic means with weight ${r\in\, ]0,1[}$ at the same time? This question is a functional equation problem: Characterize the functions ${\varphi,\psi\in CM(I)}$ and the parameters ${p,q\in\,]0,1]}$ , ${r\in\,]0,1[}$ for which the equation $$L_\varphi^{(p,q)}(x,y)=L_\psi^{(r,1-r)}(x,y)$$ holds for all ${x,y\in I}$ .     

A Short Note on Essentially Σ1 Sentences

Guaspari (J Symb Logic 48:777–789, 1983) conjectured that a modal formula is it essentially Σ₁ (i.e., it is Σ₁ under any arithmetical interpretation), if and only if it is provably equivalent to a disjunction of formulas of the form ${\square{B}}$ . This conjecture was proved first by A. Visser. Then, in (de Jongh and Pianigiani, Logic at Work: In Memory of Helena Rasiowa, Springer-Physica Verlag, Heidelberg-New York, pp. 246–255, 1999), the authors characterized essentially Σ₁ formulas of languages including witness comparisons using the interpretability logic ILM. In this note we give a similar characterization for formulas with a binary operator interpreted as interpretability in a finitely axiomatizable extension of IΔ ₀  + Supexp and we address a similar problem for IΔ ₀  + Exp.     

Normal Singular Integral Operators with Cauchy Kernel on L 2

Let α and β be functions in ${L^\infty(\mathbb{T})}$ , where ${\mathbb{T}}$ is the unit circle. Let P denote the orthogonal projection from ${L^2(\mathbb{T})}$ onto the Hardy space ${H^2(\mathbb{T})}$ , and Q = I ? P, where I is the identity operator on ${L^2(\mathbb{T})}$ . This paper is concerned with the singular integral operators S _α,β on ${L^2(\mathbb{T})}$ of the form S _α,β f = αPf + βQf, for ${f \in L^2(\mathbb{T})}$ . In this paper, we study the normality of S _α,β which is related to the Brown–Halmos theorem for the normal Toeplitz operator on ${H^2(\mathbb{T})}$ .     

Martin’s Maximum and tower forcing

There are several examples in the literature showing that compactness-like properties of a cardinal κ cause poor behavior of some generic ultrapowers which have critical point κ (Burke [1] when κ is a supercompact cardinal; Foreman-Magidor [6] when κ = ω ₂ in the presence of strong forcing axioms). We prove more instances of this phenomenon. First, the Reflection Principle (RP) implies that if $\overrightarrow I $ is a tower of ideals which concentrates on the class $GI{C_{{\omega _1}}}$ of ω ₁-guessing, internally club sets, then $\overrightarrow I $ is not presaturated (a set is ω ₁-guessing iff its transitive collapse has the ω ₁-approximation property as defined in Hamkins [10]). This theorem, combined with work from [16], shows that if PFA ⁺ or MM holds and there is an inaccessible cardinal, then there is a tower with critical point ω ₂ which is not presaturated; moreover, this tower is significantly different from the non-presaturated tower already known (by Foreman-Magidor [6]) to exist in all models of Martin’s Maximum. The conjunction of the Strong Reflection Principle (SRP) and the Tree Property at ω ₂ has similar implications for towers of ideals which concentrate on the wider class $GI{C_{{\omega _1}}}$ of ω ₁-guessing, internally stationary sets. Finally, we show that the word “presaturated” cannot be replaced by “precipitous” in the theorems above: Martin’s Maximum (which implies SRP and the Tree Property at ω ₂) is consistent with a precipitous tower on $GI{C_{{\omega _1}}}$ .     

A base-matrix lemma for sets of rationals modulo nowhere dense sets

We study some properties of the quotient forcing notions ${Q_{tr(I)} = \wp(2^{< \omega})/tr(I)}$ and P _I ?= B(2 ^ω )/I in two special cases: when I is the σ-ideal of meager sets or the σ-ideal of null sets on 2 ^ω . We show that the remainder forcing R _I =?Q _tr(I)/P _I is σ-closed in these cases. We also study the cardinal invariant of the continuum ${\mathfrak{h}_{\mathbb{Q}}}$ , the distributivity number of the quotient ${Dense(\mathbb{Q})/nwd}$ , in order to show that ${\wp(\mathbb{Q})/nwd}$ collapses ${\mathfrak{c}}$ to ${\mathfrak{h}_{\mathbb{Q}}}$ , thus answering a question addressed in Balcar et?al. (Fundamenta Mathematicae 183:59–80, 2004).     

The cardinal coefficients of the Ideal {{\mathcal {I}}_{f}}

In 2002, Yorioka introduced the σ-ideal ${{\mathcal {I}}_f}$ for strictly increasing functions f from ω into ω to analyze the cofinality of the strong measure zero ideal. For each f, we study the cardinal coefficients (the additivity, covering number, uniformity and cofinality) of ${{\mathcal {I}}_f}$ .     

Covering properties of ideals

Elekes proved that any infinite-fold cover of a σ-finite measure space by a sequence of measurable sets has a subsequence with the same property such that the set of indices of this subsequence has density zero. Applying this theorem he gave a new proof for the random-indestructibility of the density zero ideal. He asked about other variants of this theorem concerning I-almost everywhere infinite-fold covers of Polish spaces where I is a σ-ideal on the space and the set of indices of the required subsequence should be in a fixed ideal ${{\mathcal{J}}}$ on ω. We introduce the notion of the ${{\mathcal{J}}}$ -covering property of a pair ${({\mathcal{A}}, I)}$ where ${{\mathcal{A}}}$ is a σ-algebra on a set X and ${{I \subseteq \mathcal{P}(X)}}$ is an ideal. We present some counterexamples, discuss the category case and the Fubini product of the null ideal ${\mathcal{N}}$ and the meager ideal ${\mathcal{M}}$ . We investigate connections between this property and forcing-indestructibility of ideals. We show that the family of all Borel ideals ${{\mathcal{J}}}$ on ω such that ${\mathcal{M}}$ has the ${{\mathcal{J}}}$ -covering property consists exactly of non weak Q-ideals. We also study the existence of smallest elements, with respect to Katětov–Blass order, in the family of those ideals ${\mathcal{J}}$ on ω such that ${\mathcal{N}}$ or ${\mathcal{M}}$ has the ${\mathcal{J}}$ -covering property. Furthermore, we prove a general result about the cases when the covering property “strongly” fails.     

On the Generalized Feynman–Kac Transformation for?Nearly Symmetric Markov Processes

Suppose that X is a right process which is associated with a non-symmetric Dirichlet form $(\mathcal{E},D(\mathcal{E}))$ on L ²(E;m). For $u\in D(\mathcal{E})$ , we have Fukushima??s decomposition: $\tilde{u}(X_{t})-\tilde{u}(X_{0})=M^{u}_{t}+N^{u}_{t}$ . In this paper, we investigate the strong continuity of the generalized Feynman?CKac semigroup defined by $P^{u}_{t}f(x)=E_{x}[e^{N^{u}_{t}}f(X_{t})]$ . Let $Q^{u}(f,g)=\mathcal{E}(f,g)+\mathcal{E}(u,fg)$ for $f,g\in D(\mathcal{E})_{b}$ . Denote by J ₁ the dissymmetric part of the jumping measure J of $(\mathcal{E},D(\mathcal{E}))$ . Under the assumption that J ₁ is finite, we show that $(Q^{u},D(\mathcal{E})_{b})$ is lower semi-bounded if and only if there exists a constant ?? ₀??0 such that $\|P^{u}_{t}\|_{2}\leq e^{\alpha_{0}t}$ for every t>0. If one of these conditions holds, then $(P^{u}_{t})_{t\geq0}$ is strongly continuous on L ²(E;m). If X is equipped with a differential structure, then this result also holds without assuming that J ₁ is finite.     

Geometric ergodicity and the spectral gap of non-reversible Markov chains

We argue that the spectral theory of non-reversible Markov chains may often be more effectively cast within the framework of the naturally associated weighted-L _∞ space ${L_\infty^V}$ , instead of the usual Hilbert space L ₂?=?L ₂(π), where π is the invariant measure of the chain. This observation is, in part, based on the following results. A discrete-time Markov chain with values in a general state space is geometrically ergodic if and only if its transition kernel admits a spectral gap in ${L_\infty^V}$ . If the chain is reversible, the same equivalence holds with L ₂ in place of ${L_\infty^V}$ . In the absence of reversibility it fails: There are (necessarily non-reversible, geometrically ergodic) chains that admit a spectral gap in ${L_\infty^V}$ but not in L ₂. Moreover, if a chain admits a spectral gap in L ₂, then for any ${h\in L_2}$ there exists a Lyapunov function ${V_h\in L_1}$ such that V _h dominates h and the chain admits a spectral gap in ${L_\infty^{V_h}}$ . The relationship between the size of the spectral gap in ${L_\infty^V}$ or L ₂, and the rate at which the chain converges to equilibrium is also briefly discussed.     

ε_0-Regularity for Mean Curvature Flow from Surface to Flat Riemannian Manifold

In this paper we prove an ε0-regularity theorem for mean curvature flow from surface to a flat Riemannian manifold. More precisely, we prove that if the initial energy ∫Σ0 |A|2 ≤ε0 and the initial area μ0(Σ0) is not large, then along the mean curvature flow, we have ∫Σt|A|2 ≤ε0. As an application, we obtain the long time existence and convergence result of the mean curvature flow.