The ∀∃-theory of the effectively closed Medvedev degrees is decidable

We show that there is a computable procedure which, given an ??-sentence ${\varphi}$ in the language of the partially ordered sets with a top element 1 and a bottom element 0, computes whether ${\varphi}$ is true in the Medvedev degrees of ${\Pi^0_1}$ classes in Cantor space, sometimes denoted by ${\mathcal{P}_s}$ .     

Democracy functions and optimal embeddings for approximation spaces

We prove optimal embeddings for nonlinear approximation spaces $\mathcal{A}^{\alpha}_q$ , in terms of weighted Lorentz sequence spaces, with the weights depending on the democracy functions of the basis. As applications we recover known embeddings for N-term wavelet approximation in L ^p , Orlicz, and Lorentz norms. We also study the ??greedy classes?? ${\mathcal{G}_{q}^{\alpha}}$ introduced by Gribonval and Nielsen, obtaining new counterexamples which show that ${\mathcal{G}_{q}^{\alpha}}\not=\mathcal{A}^{\alpha}_q$ for most non-democratic unconditional bases.     

A Hexagonal Framework of the Field {\mathbb{F}_4} and the Associated Borromean Logic

The hexagonal structure for ??the geometry of logical opposition??, as coming from Aristoteles?CApuleius square and Sesmat?CBlanché hexagon, is presented here in connection with, on the one hand, geometrical ideas on duality on triangles (construction of ??companion??), and on the other hand, constructions of tripartitions, emphasizing that these are exactly cases of borromean objects. Then a new case of a logical interest introduced here is the double magic tripartition determining the semi-ring ${\mathcal{B}_3}$ and this is a borromean object again, in the heart of the semi-ring ${{\rm Mat}_{3}(\mathbb{B}_{\rm Alg})}$ . With this example we understand better in which sense the borromean object is a deepening of the hexagon, in a logical vein. Then, and this is our main objective here, the Post-Mal??cev full iterative algebra ${\mathbb{P}_4 = \mathbb{P}(\mathbb{F}_4)}$ of functions of all arities on ${\mathbb{F}_4}$ , is proved to be a borromean object, generated by three copies of ${\mathbb{P}_2}$ in it. This fact is induced by a hexagonal structure of the field ${\mathbb{F}_4}$ . This hexagonal structure is seen as precisely a geometrical addition to standard boolean logic, exhibiting ${\mathbb{F}_4}$ as a ??boolean manifold??. This structure allows to analyze also ${\mathbb{P}_4}$ as generated by adding to a boolean set of logical functions a very special modality, namely the Frobenius squaring map in ${\mathbb{F}_4}$ . It is related to the splitting of paradoxes, to modified logic, to specular logic. It is a setting for a theory of paradoxical sentences, seen as computations of movements on the bi-hexagonal link among the 12 classical logics on a set of 4 values.     

Composition Operators on Spaces of Fractional Cauchy Transforms

For ?? > 0, the Banach space ${\mathcal{F}_{\alpha}}$ is defined as the collection of functions f which can be represented as integral transforms of an appropriate kernel against a Borel measure defined on the unit circle T. Let ?? be an analytic self-map of the unit disc D. The map ?? induces a composition operator on ${\mathcal{F}_{\alpha}}$ if ${C_{\Phi}(f) = f \circ \Phi \in \mathcal{F}_{\alpha}}$ for any function ${f \in \mathcal{F}_{\alpha}}$ . Various conditions on ?? are given, sufficient to imply that C _?? is bounded on ${\mathcal{F}_{\alpha}}$ , in the case 0 < ?? < 1. Several of the conditions involve ???? and the theory of multipliers of the space ${\mathcal{F}_{\alpha}}$ . Relations are found between the behavior of C _?? and the membership of ?? in the Dirichlet spaces. Conditions given in terms of the generalized Nevanlinna counting function are shown to imply that ?? induces a bounded composition operator on ${\mathcal{F}_{\alpha}}$ , in the case 1/2 ?? ?? < 1. For such ??, examples are constructed such that ${\| \Phi \|_{\infty} = 1}$ and ${C_{\Phi}: \mathcal{F}_{\alpha} \rightarrow \mathcal{F}_{\alpha}}$ is bounded.     

Lower large deviations for the maximal flow through a domain of {\mathbb{R}^d} in first passage percolation

We consider the standard first passage percolation model in the rescaled graph ${\mathbb{Z}^d/n}$ for d??? 2, and a domain ?? of boundary ?? in ${\mathbb{R}^d}$ . Let ??¹ and ??² be two disjoint open subsets of ??, representing the parts of ?? through which some water can enter and escape from ??. We investigate the asymptotic behaviour of the flow ${\phi_n}$ through a discrete version ?? _n of ?? between the corresponding discrete sets ${\Gamma^{1}_{n}}$ and ${\Gamma^{2}_{n}}$ . We prove that under some conditions on the regularity of the domain and on the law of the capacity of the edges, the lower large deviations of ${\phi_n/ n^{d-1}}$ below a certain constant are of surface order.     

Pointwise Convergence of the Calderón Reproducing Formula

In this paper, we study the pointwise convergence of the Calderón reproducing formula, which is also known as an inversion formula for wavelet transforms. We show that for every $f\in L_{w}^{p}(\mathbb {R}^{d})$ with an $\mathcal{A}_{p}$ weight w, 1??p<??, the integral is convergent at every Lebesgue point of f, and therefore almost everywhere. Moreover, we prove the convergence without any assumption on the smoothness of wavelet functions.     

Logarithmic Mean Oscillation on the Polydisc,Multi-Parameter Paraproducts and Iterated Commutators

We introduce another notion of bounded logarithmic mean oscillation in the \(N\) -torus and give an equivalent definition in terms of boundedness of multi-parameter paraproducts from the dyadic little \(\mathrm {BMO}\) , \(\mathrm {bmo}^d(\mathbb {T}^N)\) to the dyadic product \(\mathrm {BMO}\) space, \(\mathrm {BMO}^d(\mathbb {T}^N)\) . We also obtain a sufficient condition for the boundedness of the iterated commutators from the subspace of \(\mathrm {bmo}(\mathbb {R}^N)\) consisting of functions with support in \([0,1]^N\) to \(\mathrm {BMO}(\mathbb {R}^N)\) .     

Isolated boundary singularities of semilinear elliptic equations

Given a smooth domain ${\Omega\subset\mathbb{R}^N}$ such that ${0 \in \partial\Omega}$ and given a nonnegative smooth function ?? on ???, we study the behavior near 0 of positive solutions of ???u?=?u ^q in ?? such that u =? ?? on ???\{0}. We prove that if ${\frac{N+1}{N-1} < q < \frac{N+2}{N-2}}$ , then ${u(x)\leq C |x|^{-\frac{2}{q-1}}}$ and we compute the limit of ${|x|^{\frac{2}{q-1}} u(x)}$ as x ?? 0. We also investigate the case ${q= \frac{N+1}{N-1}}$ . The proofs rely on the existence and uniqueness of solutions of related equations on spherical domains.     

Approximate dual Gabor atoms via the adjoint lattice method

Regular Gabor frames for \({\boldsymbol {L}{^{2}}(\mathbb {R}^d)}\) are obtained by applying time-frequency shifts from a lattice in \(\boldsymbol {\Lambda } \vartriangleleft {\mathbb {R}^{d} \times \mathbb {\widehat {R}}}\) to some decent so-called Gabor atom g, which typically is something like a summability kernel in classical analysis, or a Schwartz function, or more generally some \(g \in {\boldsymbol {S}_{0}(\mathbb {R}^{d})}\) . There is always a canonical dual frame, generated by the dual Gabor atom \({\widetilde g}\) . The paper promotes a numerical approach for the efficient calculation of good approximations to the dual Gabor atom for general lattices, including the non-separable ones (different from \({a\mathbb {Z}^{d}\,{\times }\,b\mathbb {Z}^{d}}\) ). The theoretical foundation for the approach is the well-known Wexler-Raz biorthogonality relation and the more recent theory of localized frames. The combination of these principles guarantees that the dual Gabor atom can be approximated by a linear combination of a few time-frequency shifted atoms from the adjoint lattice \(\boldsymbol {\Lambda }\circ\) . The effectiveness of this approach is justified by a new theoretical argument and demonstrated by numerical examples.     

Band-limited Wavelets and Framelets in Low Dimensions

In this paper, we study the problem of constructing non-separable band-limited wavelet tight frames, Riesz wavelets and orthonormal wavelets in $\mathbb {R}^{2}$ and $\mathbb {R}^{3}$ . We first construct a class of non-separable band-limited refinable functions in low-dimensional Euclidean spaces by using univariate Meyer’s refinable functions along multiple directions defined by classical box-spline direction matrices. These non-separable band-limited definable functions are then used to construct non-separable band-limited wavelet tight frames via the unitary and oblique extension principles. However, these refinable functions cannot be used for constructing Riesz wavelets and orthonormal wavelets in low dimensions as they are not stable. Another construction scheme is then developed to construct stable refinable functions in low dimensions by using a special class of direction matrices. The resulting stable refinable functions allow us to construct a class of MRA-based non-separable band-limited Riesz wavelets and particularly band-limited orthonormal wavelets in low dimensions with small frequency support.     

Comparison of two regulatory results on functional equations with few parameters

Two methods to prove regularity properties of the linear functional equation $$f(x)=h_0(x,y)+\sum_{j=1}^n h_j(x,y)f(x+g_j(y)), $$ where ${(x,y) \in D \subset \mathbb{R}^r \times \mathbb{R}^s}$ , ${x \in \mathbb{R}^r}$ and ${y \in \mathbb{R}^s}$ , with few parameters i.e. allowing 1 ?? s < r are examined. It is proved that??under certain conditions, for some class of equations and in some sense??they are equivalent.     

Ground state and multiple solutions for a critical exponent problem

We study the following Brezis?CNirenberg type critical exponent equation which is related to the Yamabe problem: $$-\Delta u=\lambda u+ |u|^{2^{\ast}-2}u, \quad u\in H_0^1 (\Omega),$$ where ?? is a smooth bounded domain in ${{\mathbb R}^N(N\ge3)}$ and 2^* is the critical Sobolev exponent. We show that, if N ?? 5, this problem has at least ${\lceil\frac{N+1}{2}\rceil}$ pairs of nontrivial solutions for each fixed ?? ?? ??₁, where ??₁ is the first eigenvalue of ??? with the Dirichlet boundary condition. For N ?? 3, we give energy estimates from below for ground state solutions.     

Sharp inequalities for dyadic A 1 weights

We show how the Bellman function method can be used to obtain sharp inequalities for the maximal operator of a dyadic A ₁ weight on ${\mathbb{R}^n}$ R n . Using this approach, we determine the optimal constants in the corresponding weak-type estimates. Furthermore, we provide an alternative, simpler proof of the related maximal L ^p -inequalities, originally shown by Melas.     

Surfaces of Rotation with Constant Extrinsic Curvature in a Conformally Flat 3-Space

In this work, we relate the extrinsic curvature of surfaces with respect to the Euclidean metric and any metrics that are conformal to the Euclidean metric. We introduce the space ${\mathbb{E}_3}$ ??the 3-dimensional real vector space equipped with a conformally flat metric that is a solution of the Einstein equation. We characterize the surfaces of rotation with constant extrinsic curvature in the space ${\mathbb{E}_3}$ . We obtain a one-parameter family of two-sheeted hyperboloids that are complete surfaces with zero extrinsic curvature in ${\mathbb{E}_3}$ . Moreover, we obtain a one-parameter family of cones and show that there exists another one-parameter family of complete surfaces of rotation with zero extrinsic curvature in ${\mathbb{E}_3}$ . Moreover, we show that there exist complete surfaces with constant negative extrinsic curvature in ${\mathbb{E}_3}$ . As an application we prove that there exist complete surfaces with Gaussian curvature K ?? ? ?? < 0, in contrast with Efimov??s Theorem for the Euclidean space, and Schlenker??s Theorem for the hyperbolic space.     

Zhelobenko invariants, Bernstein-Gelfand-Gelfand operators and the analogue Kostant Clifford algebra conjecture

Let $ \mathfrak{g} $ be a complex simple Lie algebra and $ \mathfrak{h} $ a Cartan subalgebra. The Clifford algebra C( $ \mathfrak{g} $ ) of g admits a Harish-Chandra map. Kostant conjectured (as communicated to Bazlov in about 1997) that the value of this map on a (suitably chosen) fundamental invariant of degree 2?m?+?1 is just the zero weight vector of the simple (2?m?+?1)-dimensional module of the principal s-triple obtained from the Langlands dual $ {\mathfrak{g}^\vee } $ . Bazlov [1] settled this conjecture positively in type A. The hard part of the Kostant Clifford algebra conjecture is a question concerning the Harish-Chandra map for the enveloping algebra U( $ \mathfrak{g} $ ) composed with evaluation at the half sum ?? of the positive roots. The analogue Kostant conjecture is obtained by replacing the Harish-Chandra map by a ??generalized Harish-Chandra?? map. This map had been studied notably by Zhelobenko [15]. The proof given here involves a symmetric algebra version of the Kostant conjecture, the Zhelobenko invariants in the adjoint case, and, surprisingly, the Bernstein-Gelfand-Gelfand operators introduced in their study [3] of the cohomology of the flag variety.     

Linearity and Second Fundamental Forms for Proper Holomorphic Maps from \mathbb{B}^{n+1} to \mathbb{B}^{4n-3}

For a holomorphic proper map F from the ball $\mathbb{B}^{n+1}$ into $\mathbb{B}^{N+1}$ that is C ³ smooth up to the boundary, the image $M=F(\partial\mathbb{B}^{n})$ is an immersed CR submanifold in the sphere $\partial \mathbb{B}^{N+1}$ on which some second fundamental forms II _M and $\mathit{II}^{CR}_{M}$ can be defined. It is shown that when 4??n+1<N+1??4n?3, F is linear fractional if and only if $\mathit{II}_{M} - \mathit{II}_{M}^{CR} \equiv 0$ .     

Differentiation and the Balian-Low Theorem

The Balian-Low theorem (BLT) is a key result in time-frequency analysis, originally stated by Balian and, independently, by Low, as: If a Gabor system $\{e^{2\pi imbt} \, g(t-na)\}_{m,n \in \mbox{\bf Z}}$ with $ab=1$ forms an orthonormal basis for $L^2({\bf R}),$ then $\left(\int_{-\infty}^\infty |t \, g(t)|^2 \, dt\right) \, \left(\int_{-\infty}^\infty |\gamma \, \hat g(\gamma)|^2 \, d\gamma\right) = +\infty.$ The BLT was later extended from orthonormal bases to exact frames. This paper presents a tutorial on Gabor systems, the BLT, and related topics, such as the Zak transform and Wilson bases. Because of the fact that $(g')^{\wedge}(\gamma) = 2 \pi i \gamma \, \hat g(\gamma)$ , the role of differentiation in the proof of the BLT is examined carefully. The major new contributions of this paper are the construction of a complete Gabor system of the form $\{e^{2\pi ib_mt\} \, g(t-a_n)}$ such that $\{(a_n,b_m)\}$ has density strictly less than 1, an Amalgam BLT that provides distinct restrictions on Gabor systems $\{e^{2\pi imbt} \, g(t-na)\}$ that form exact frames, and a new proof of the BLT for exact frames that does not require differentiation and relies only on classical real variable methods from harmonic analysis.     

On convex relaxations for quadratically constrained quadratic programming

We consider convex relaxations for the problem of minimizing a (possibly nonconvex) quadratic objective subject to linear and (possibly nonconvex) quadratic constraints. Let $\mathcal{F }$ denote the feasible region for the linear constraints. We first show that replacing the quadratic objective and constraint functions with their convex lower envelopes on $\mathcal{F }$ is dominated by an alternative methodology based on convexifying the range of the quadratic form $\genfrac(){0.0pt}{}{1}{x}\genfrac(){0.0pt}{}{1}{x}^T$ for $x\in \mathcal{F }$ . We next show that the use of ?? $\alpha $ BB?? underestimators as computable estimates of convex lower envelopes is dominated by a relaxation of the convex hull of the quadratic form that imposes semidefiniteness and linear constraints on diagonal terms. Finally, we show that the use of a large class of D.C. (??difference of convex??) underestimators is dominated by a relaxation that combines semidefiniteness with RLT constraints.     

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.     

Spinor Bishop Equations of Curves in Euclidean 3-Space

In this paper, we study spinor Bishop equations of curves in ${\mathbb{E}^3}$ . We research the spinor formulations of curves according to Bishop frames in ${\mathbb{E}^3}$ . Also, the relations between spinor formulations of Bishop frames and Frenet frame are expressed.