Sharp weighted norm inequalities for Littlewood–Paley operators and singular integrals

We prove sharp Lp(w) norm inequalities for the intrinsic square function (introduced recently by M. Wilson) in terms of the Ap characteristic of w for all 1<p<∞. This implies the same sharp inequalities for the classical Lusin area integral S(f), the Littlewood–Paley g-function, and their continuous analogs Sψ and gψ. Also, as a corollary, we obtain sharp weighted inequalities for any convolution Calderón–Zygmund operator for all 1<p?3/2 and 3?p<∞, and for its maximal truncations for 3?p<∞.     

Unification in linear temporal logic LTL

We prove that a propositional Linear Temporal Logic with Until and Next (LTL) has unitary unification. Moreover, for every unifiable in LTL formula A there is a most general projective unifier, corresponding to some projective formula B, such that A is derivable from B in LTL. On the other hand, it can be shown that not every open and unifiable in LTL formula is projective. We also present an algorithm for constructing a most general unifier.     

Pinching theorems of hypersurfaces in a unit sphere

Let Mn be a complete hypersurface in Sn+1(1) with constant mean curvature. Assume that Mn has n−1 principal curvatures with the same sign everywhere. We prove that if RicM≤C₋(H), either S?S₊(H) or RicM?0 or the fundamental group of Mn is infinite, then S is constant, S=S₊(H) and Mn is isometric to a Clifford torus with . These rigidity theorems are still valid for compact hypersurface without constancy condition on the mean curvature.     

BLO spaces associated with the Ornstein–Uhlenbeck operator

Let (Rn,|⋅|,dγ) be the Gauss measure metric space, where Rn denotes the n-dimensional Euclidean space, |⋅| the Euclidean norm and for all x∈Rn the Gauss measure. In this paper, for any a∈(0,∞), the authors introduce some BLOa(γ) space, namely, the space of functions with bounded lower oscillation associated with a given class of admissible balls with parameter a. Then the authors prove that the noncentered local natural Hardy–Littlewood maximal operator is bounded from BMO(γ) of Mauceri and Meda to BLOa(γ). Moreover, a characterization of the space BLOa(γ), via the local natural maximal operator and BMO(γ), is given. The authors further prove that a class of maximal singular integrals, including the corresponding maximal operators of both imaginary powers of the Ornstein–Uhlenbeck operator and Riesz transforms of any order associated with the Ornstein–Uhlenbeck operator, are bounded from L^∞(γ) to BLOa(γ).     

CERES in higher-order logic

We define a generalization of the first-order cut-elimination method CERES to higher-order logic. At the core of lies the computation of an (unsatisfiable) set of sequents (the characteristic sequent set) from a proof π of a sequent S. A refutation of in a higher-order resolution calculus can be used to transform cut-free parts of π (the proof projections) into a cut-free proof of S. An example illustrates the method and shows that can produce meaningful cut-free proofs in mathematics that traditional cut-elimination methods cannot reach.     

Independence results for variants of sharply bounded induction

The theory , axiomatized by the induction scheme for sharply bounded formulae in Buss’ original language of bounded arithmetic (with ⌊x/2⌋ but not ⌊x/2^y⌋), has recently been unconditionally separated from full bounded arithmetic S₂. The method used to prove the separation is reminiscent of those known from the study of open induction.We make the connection to open induction explicit, showing that models of can be built using a “nonstandard variant” of Wilkie’s well-known technique for building models of IOpen. This makes it possible to transfer many results and methods from open to sharply bounded induction with relative ease.We provide two applications: (i) the Shepherdson model of IOpen can be embedded into a model of , which immediately implies some independence results for ; (ii) extended by an axiom which roughly states that every number has a least 1 bit in its binary notation, while significantly stronger than plain , does not prove the infinity of primes.     

Modulation spaces, Wiener amalgam spaces, and Brownian motions

We study the local-in-time regularity of the Brownian motion with respect to localized variants of modulation spaces and Wiener amalgam spaces . We show that the periodic Brownian motion belongs locally in time to and for (s−1)q<−1, and the condition on the indices is optimal. Moreover, with the Wiener measure μ on T, we show that and form abstract Wiener spaces for the same range of indices, yielding large deviation estimates. We also establish the endpoint regularity of the periodic Brownian motion with respect to a Besov-type space . Specifically, we prove that the Brownian motion belongs to for (s−1)p=−1, and it obeys a large deviation estimate. Finally, we revisit the regularity of Brownian motion on usual local Besov spaces , and indicate the endpoint large deviation estimates.     

A characterization of the Language Invariant families satisfying Spectrum Exchangeability in Polyadic Inductive Logic

A necessary and sufficient condition in terms of a de Finetti style representation is given for a probability function in Polyadic Inductive Logic to satisfy being part of a Language Invariant family satisfying Spectrum Exchangeability. This theorem is then considered in relation to the unary Carnap and Nix-Paris Continua.     

An algebraic slice in the coadjoint space of the Borel and the Coxeter element

Let g be a complex simple Lie algebra and b a Borel subalgebra. The algebra Y of polynomial semi-invariants on the dual b^? of b is a polynomial algebra on rank g generators (Grothendieck and Dieudonné (1965–1967)) [16]. The analogy with the semisimple case suggests there exists an algebraic slice to coadjoint action, that is an affine translate y+V of a vector subspace of b^? such that the restriction map induces an isomorphism of Y onto the algebra R[y+V] of regular functions on y+V. This holds in type A and even extends to all biparabolic subalgebras (Joseph (2007)) [20]; but the construction fails in general even with respect to the Borel. Moreover already in type C(2) no algebraic slice exists.Very surprisingly the exception of type C(2) is itself an exception. Indeed an algebraic slice for the coadjoint action of the Borel subalgebra is constructed for all simple Lie algebras except those of types B(2m), C(n) and F(4).Outside type A, the slice obtained meets an open dense subset of regular orbits, even though the special point y of the slice is not itself regular. This explains the failure of our previous construction.     

On spacelike hypersurfaces with constant scalar curvature in the anti-de Sitter space

In this paper, we classify complete spacelike hypersurfaces in the anti-de Sitter space (n?3) with constant scalar curvature and with two principal curvatures. Moreover, we prove that if Mn is a complete spacelike hypersurface with constant scalar curvature n(n−1)R and with two distinct principal curvatures such that the multiplicity of one of the principal curvatures is n−1, then R<(n−2)c/n. Additionally, we also obtain several rigidity theorems for such hypersurfaces.     

On duality and fractionality of multicommodity flows in directed networks

In this paper we address a topological approach to multiflow (multicommodity flow) problems in directed networks. Given a terminal weight μ, we define a metrized polyhedral complex, called the directed tight span T_μ, and prove that the dual of the μ-weighted maximum multiflow problem reduces to a facility location problem on T_μ. Also, in case where the network is Eulerian, it further reduces to a facility location problem on the tropical polytope spanned by μ. By utilizing this duality, we establish the classifications of terminal weights admitting a combinatorial min–max relation (i) for every network and (ii) for every Eulerian network. Our result includes the Lomonosov–Frank theorem for directed free multiflows and Ibaraki–Karzanov–Nagamochi’s directed multiflow locking theorem as special cases.     

On the Beurling dimension of exponential frames

We study Fourier frames of exponentials on fractal measures associated with a class of affine iterated function systems. We prove that, under a mild technical condition, the Beurling dimension of a Fourier frame coincides with the Hausdorff dimension of the fractal.     

The algebraic structure of relative twisted vertex operators

Twisted vertex operators based on rational lattices have had many applications in vertex operator algebra theory and conformal field theory. In this paper, “relativized” twisted vertex operators are constructed in a general context based on isometries of rational lattices, and a generalized twisted Jacobi identity is established for them. This result generalizes many previous results. Relatived untwisted vertex operators had been studied in a monograph by the authors. The present paper includes as a special case the proof of the main relations among twisted vertex operators based on even lattices announced some time ago by the second author.     

Rank properties of subspaces of symmetric and hermitian matrices over finite fields

We investigate constant rank subspaces of symmetric and hermitian matrices over finite fields, using a double counting method related to the number of common zeros of the corresponding subspaces of symmetric bilinear and hermitian forms. We obtain optimal bounds for the dimensions of constant rank subspaces of hermitian matrices, and good bounds for the dimensions of subspaces of symmetric and hermitian matrices whose non-zero elements all have odd rank.     

Unconditional basic sequences in spaces of large density

We study the problem of the existence of unconditional basic sequences in Banach spaces of high density. We show in particular the relative consistency of the statement that every Banach space of density ℵω contains an unconditional basic sequence.     

Conformal fields and the stability of leaves with constant higher order mean curvature

In this paper, we study hypersurfaces with constant rth mean curvature Sr. We investigate the stability of such hypersurfaces in the case when they are leaves of a codimension one foliation. We also generalize recent results by Barros and Sousa, concerning conformal fields, to an arbitrary manifold. Using this we show that normal component of a Killing field is an rth Jacobi field of a hypersurface with Sr+1 constant. Finally, we study relations between rth Jacobi fields and vector fields preserving a foliation.     

Optimal inverse Littlewood–Offord theorems

Let ηi, i=1,…,n, be iid Bernoulli random variables, taking values ±1 with probability . Given a multiset V of n integers v₁,…,vn, we define the concentration probability as A classical result of Littlewood–Offord and Erd?s from the 1940s asserts that, if the vi are non-zero, then ρ(V) is O(n−1/2). Since then, many researchers have obtained improved bounds by assuming various extra restrictions on V.About 5 years ago, motivated by problems concerning random matrices, Tao and Vu introduced the inverse Littlewood–Offord problem. In the inverse problem, one would like to characterize the set V, given that ρ(V) is relatively large.In this paper, we introduce a new method to attack the inverse problem. As an application, we strengthen the previous result of Tao and Vu, obtaining an optimal characterization for V. This immediately implies several classical theorems, such as those of Sárközy and Szemerédi and Halász.The method also applies to the continuous setting and leads to a simple proof for the β-net theorem of Tao and Vu, which plays a key role in their recent studies of random matrices.All results extend to the general case when V is a subset of an abelian torsion-free group, and ηi are independent variables satisfying some weak conditions.     

Did Egyptian scribes have an algorithmic means for determining the circumference of a circle?

It has been at various times proposed in regard to Problem 10 of the Moscow Mathematical Papyrus that Egyptian scribes had developed a computational algorithm by which they could calculate the circumference of a circle of known diameter length. It has also been proposed at various times that the 7:22 ratio between a circle’s diameter and its circumference was known by the Egyptian scribes at a surprisingly early period.The present paper explores currently available evidence that lends support to each of these propositions.     

The Logic of Uncertain Justifications

In Artemov?s Justification Logic, one can make statements interpreted as “t is evidence for the truth of formula F.” We propose a variant of this logic in which one can say “I have degree r of confidence that t is evidence for the truth of formula F.” After defining both an axiomatic approach and a semantics for this Logic of Uncertain Justifications, we will prove the usual soundness and completeness theorems.     

Orthogonal exponentials on the generalized plane Sierpinski gasket

The self-affine measure μ_Mp,D corresponding tois supported on the the generalized plane Sierpinski gasket T(M_p,D). In the present paper we show that there exist at most 3 mutually orthogonal exponential functions in L²(μ_Mp,D), and the number 3 is the best. This generalizes several known results on the non-spectral self-affine measure problem.