Estimates in Beurling-Helson type theorems: Multidimensional case

We consider the spaces A _p ( $\mathbb{T}^m $ ) of functions f on the m-dimensional torus $\mathbb{T}^m $ such that the sequence of Fourier coefficients $\hat f = \{ \hat f(k),k \in \mathbb{Z}^m \} $ belongs to l ^p (? ^m ), 1 ≤ p < 2. The norm on A _p ( $\mathbb{T}^m $ ) is defined by $\left\| f \right\|_{A_p (\mathbb{T}^m )} = \left\| {\hat f} \right\|_{l^p (\mathbb{Z}^m )} $ . We study the rate of growth of the norms $\left\| {e^{i\lambda \phi } } \right\|_{A_p (\mathbb{T}^m )} $ as |λ| → ∞, λ ∈ ?, for C ¹-smooth real functions φ on $\mathbb{T}^m $ (the one-dimensional case was investigated by the author earlier). The lower estimates that we obtain have direct analogs for the spaces A _p (? ^m ).     

Hard Sard: Quantitative Implicit Function and Extension Theorems for Lipschitz Maps

We prove a global implicit function theorem. In particular we show that any Lipschitz map ${f : \mathbb{R}^{n} \times \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}}$ (with n-dim. image) can be precomposed with a bi-Lipschitz map ${\bar{g} : \mathbb{R}^{n} \times \mathbb{R}^{m} \rightarrow \mathbb{R}^{n} \times \mathbb{R}^{m}}$ such that ${f \circ \bar{g}}$ will satisfy, when we restrict to a large portion of the domain ${E \subset \mathbb{R}^{n} \times \mathbb{R}^{m}}$ , that ${f \circ \bar{g}}$ is bi-Lipschitz in the first coordinate, and constant in the second coordinate. Geometrically speaking, the map ${\bar{g}}$ distorts ${\mathbb{R}^{n+m}}$ in a controlled manner so that the fibers of f are straightened out. Furthermore, our results stay valid when the target space is replaced by any metric space. A main point is that our results are quantitative: the size of the set E on which behavior is good is a significant part of the discussion. Our estimates are motivated by examples such as Kaufman’s 1979 construction of a C ¹ map from [0, 1]³ onto [0, 1]² with rank ≤ 1 everywhere. On route we prove an extension theorem which is of independent interest. We show that for any D ≥ n, any Lipschitz function ${f : [0,1]^{n} \rightarrow \mathbb{R}^{D}}$ gives rise to a large (in an appropriate sense) subset ${E \subset [0,1]^{n}}$ such that ${f|_E}$ is bi-Lipschitz and may be extended to a bi-Lipschitz function defined on all of ${\mathbb{R}^{n}}$ . This extends results of Jones and David, from 1988. As a simple corollary, we show that n-dimensional Ahlfors–David regular spaces lying in ${\mathbb{R}^{D}}$ having big pieces of bi-Lipschitz images also have big pieces of big pieces of Lipschitz graphs in ${\mathbb{R}^{D}}$ . This was previously known only for D ≥ 2n?+?1 by a result of David and Semmes.     

Isomorphic chain complexes of Hamiltonian dynamics on tori

In this work, for a given smooth, generic Hamiltonian ${H : \mathbb{S}^{1} \times \mathbb{T}^{2n} \rightarrow \mathbb{R}}$ on the torus ${\mathbb{T}^{2n} = \mathbb{R}^{2n}/\mathbb{Z}^{2n}}$ we construct a chain isomorphism ${\Phi_{*} : (C_{*}(H), \partial^{M}_{*}) \rightarrow (C_{*}(H), \partial^{F}_{*})}$ between the Morse complex of the Hamiltonian action AH on the free loop space of the torus ${\Lambda_{0}(\mathbb{T}^{2n})}$ and the Floer complex. Though both complexes are generated by the critical points of A _H , their boundary operators differ. Therefore, the construction of ${\Phi}$ is based on counting the moduli spaces of hybrid-type solutions which involves stating a new non-Lagrangian boundary value problem for Cauchy–Riemann type operators not yet studied in Floer theory. We finally want to note that the problem is completely symmetric. So we also could construct an isomorphism ${\Psi_{*} : (C_{*}(H), \partial^{F}_{*}) \rightarrow (C_{*}(H), \partial^{M}_{*})}$ .     

Cubic symmetric polynomials yielding variations of the Erdős–Ginzburg–Ziv theorem

Let p be a prime and let $\varphi\in\mathbb{Z}_{p}[x_{1},x_{2},\ldots, x_{p}]$ be a symmetric polynomial, where  $\mathbb {Z}_{p}$ is the field of p elements. A sequence T in  $\mathbb {Z}_{p}$ of length p is called a φ-zero sequence if φ(T)=0; a sequence in $\mathbb {Z}_{p}$ is called a φ-zero free sequence if it does not contain any φ-zero subsequence. Motivated by the EGZ theorem for the prime p, we consider symmetric polynomials $\varphi\in \mathbb {Z}_{p}[x_{1},x_{2},\ldots, x_{p}]$ , which satisfy the following two conditions: (i) every sequence in  $\mathbb {Z}_{p}$ of length 2p?1 contains a φ-zero subsequence, and (ii) the φ-zero free sequences in  $\mathbb {Z}_{p}$ of maximal length are all those containing exactly two distinct elements, where each element appears p?1 times. In this paper, we determine all symmetric polynomials in $\mathbb {Z}_{p}[x_{1},x_{2},\ldots, x_{p}]$ of degree not exceeding 3 satisfying the conditions above.     

A contribution to stability theory for nonlinear Neumann problem

In this paper we study the stability of the solutions of some nonlinear Neumann problems, under perturbations of the domains in the Hausdorff complementary topology. We consider the problem $${{\left\{\begin{array}{c}-\text{ div}\;\left(a\left( x,\nabla u_{\Omega}\right)\right)=0 \;\text{in}\; \Omega \\ {a\left( x, \nabla u_{\Omega}\right) \cdot \nu=0\; \text{on}\; \partial\Omega}\end{array}\right.}}$$ where ${{\mathbf{R}^n \times \mathbf{R}^n \rightarrow \mathbf{R}^n}}$ is a Caratheodory function satisfying the standard monotonicity and growth conditions of order p, 1?<?p?<???. If ?? _h is a uniformly bounded sequence of connected open sets in R ⁿ , n ??? 2, we prove that if ${{\Omega_{h}^{c} \rightarrow \Omega^{c}}}$ in the Hausdorff metric, ${|\Omega_{h}| \rightarrow |\Omega|}$ and the geodetic distances satisfy the inequality ${d_{\Omega}\left( x,y\right) \leq \liminf_{h} d_{\Omega_{h}} \left( x,y\right)}$ for every ${x, y \in \Omega,}$ then ${\nabla u_{\Omega_h} \rightarrow\nabla u_{\Omega}}$ strongly in L ^p , provided that W ^{1, ??}(??) is dense in the space L ^{1, p} (??) of all functions whose gradient belongs to L ^p (??, R ⁿ ).     

Representation of cyclotomic fields and their subfields

Let $\mathbb{K}$ be a finite extension of a characteristic zero field $\mathbb{F}$ . We say that a pair of n × n matrices (A,B) over $\mathbb{F}$ represents $\mathbb{K}$ if $\mathbb{K} \cong {{\mathbb{F}\left[ A \right]} \mathord{\left/ {\vphantom {{\mathbb{F}\left[ A \right]} {\left\langle B \right\rangle }}} \right. \kern-0em} {\left\langle B \right\rangle }}$ , where $\mathbb{F}\left[ A \right]$ denotes the subalgebra of $\mathbb{M}_n \left( \mathbb{F} \right)$ containing A and 〈B〉 is an ideal in $\mathbb{F}\left[ A \right]$ , generated by B. In particular, A is said to represent the field $\mathbb{K}$ if there exists an irreducible polynomial $q\left( x \right) \in \mathbb{F}\left[ x \right]$ which divides the minimal polynomial of A and $\mathbb{K} \cong {{\mathbb{F}\left[ A \right]} \mathord{\left/ {\vphantom {{\mathbb{F}\left[ A \right]} {\left\langle {q\left( A \right)} \right\rangle }}} \right. \kern-0em} {\left\langle {q\left( A \right)} \right\rangle }}$ . In this paper, we identify the smallest order circulant matrix representation for any subfield of a cyclotomic field. Furthermore, if p is a prime and $\mathbb{K}$ is a subfield of the p-th cyclotomic field, then we obtain a zero-one circulant matrix A of size p × p such that (A, J) represents $\mathbb{K}$ , where J is the matrix with all entries 1. In case, the integer n has at most two distinct prime factors, we find the smallest order 0, 1-companion matrix that represents the n-th cyclotomic field. We also find bounds on the size of such companion matrices when n has more than two prime factors.     

Permanent Versus Determinant over a Finite Field

Let $ \mathbb{F} $ be a finite field of characteristic different from 2. We study the cardinality of sets of matrices with a given determinant or a given permanent for the set of Hermitian matrices $ {{\mathcal{H}}_n}\left( \mathbb{F} \right) $ and for the whole matrix space M _n ( $ \mathbb{F} $ ). It is known that for n = 2, there are bijective linear maps Φ on $ {{\mathcal{H}}_n}\left( \mathbb{F} \right) $ and M _n ( $ \mathbb{F} $ ) satisfying the condition per A = det Φ(A). As an application of the obtained results, we show that if n ≥ 3, then the situation is completely different and already for n = 3, there is no pair of maps (Φ, ?), where Φ is an arbitrary bijective map on matrices and ? : $ \mathbb{F} $ → $ \mathbb{F} $ is an arbitrary map such that per A = ?(det Φ(A)) for all matrices A from the spaces $ {{\mathcal{H}}_n}\left( \mathbb{F} \right) $ and M _n ( $ \mathbb{F} $ ), respectively. Moreover, for the space M _n ( $ \mathbb{F} $ ), we show that such a pair of transformations does not exist also for an arbitrary n > 3 if the field $ \mathbb{F} $ contains sufficiently many elements (depending on n). Our results are illustrated by a number of examples.     

On Rational Pairings of Functors

In the theory of coalgebras C over a ring R, the rational functor relates the category $_{C^*}{\mathbb{M}}$ of modules over the algebra C ^* (with convolution product) with the category $^C{\mathbb{M}}$ of comodules over C. This is based on the pairing of the algebra C ^* with the coalgebra C provided by the evaluation map ${\rm ev}:C^*\otimes_R C\to R$ . The (rationality) condition under consideration ensures that $^C{\mathbb{M}}$ becomes a coreflective full subcategory of $_{C^*}{\mathbb{M}}$ . We generalise this situation by defining a pairing between endofunctors T and G on any category ${\mathbb{A}}$ as a map, natural in $a,b\in {\mathbb{A}}$ , $$ \beta_{a,b}:{\mathbb{A}}(a, G(b)) \to {\mathbb{A}}(T(a),b), $$ and we call it rational if these all are injective. In case T?=?(T, m _T , e _T ) is a monad and G?=?(G, δ _G , ε _G ) is a comonad on ${\mathbb{A}}$ , additional compatibility conditions are imposed on a pairing between T and G. If such a pairing is given and is rational, and T has a right adjoint monad T ^???, we construct a rational functor as the functor-part of an idempotent comonad on the T-modules ${\mathbb{A}}_{T}$ which generalises the crucial properties of the rational functor for coalgebras. As a special case we consider pairings on monoidal categories.     

The Equivalence Between Seven Classes of Wavelet Multipliers and Arcwise Connectivity They Induce

Let A be a d×d expansive matrix with |detA|=2. An A-wavelet is a function $\psi\in L^{2}(\mathbb{R}^{d})$ such that $\{2^{\frac{j}{2}}\psi(A\cdot-k):\,j\in \mathbb{Z},\,k\in \mathbb{Z}^{d}\}$ is an orthonormal basis for $L^{2}(\mathbb{R}^{d})$ . A measurable function f is called an A-wavelet multiplier if the inverse Fourier transform of $f\hat{\psi}$ is an A-wavelet whenever ψ is an A-wavelet, where $\hat{\psi}$ denotes the Fourier transform of ψ. A-scaling function multiplier, A-PFW multiplier, semi-orthogonal A-PFW multiplier, MRA A-wavelet multiplier, MRA A-PFW multiplier and semi-orthogonal MRA A-PFW multiplier are defined similarly. In this paper, we prove that the above seven classes of multipliers are equivalent, and obtain a characterization of them. We then prove that if the set of all A-wavelet multipliers acts on some A-scaling function (A-wavelet, A-PFW, semi-orthogonal A-PFW, MRA A-wavelet, MRA A-PFW, semi-orthogonal MRA A-PFW), the orbit is arcwise connected in $L^{2}(\mathbb{R}^{d})$ , and that if the generator of an orbit is an MRA A-PFW, the orbit is equal to the set of all MRA A-PFWs whose Fourier transforms have same module, and is also equal to the set of all MRA A-PFWs with corresponding pseudo-scaling functions having the same module of their Fourier transforms.     

Weak Compactness and Variational Characterization of the Convexity

We prove that if X is a Banach space and ${f : X \rightarrow \mathbb{R} \cup \{+\infty\}}$ is a proper function such that f ? ? attains its minimum for every ? ε X ^*, then the sublevels of f are all relatively weakly compact in X. As a consequence we show that a Banach space X where there exists a function ${f : X \rightarrow \mathbb{R}}$ such that f ? ? attains its minimum for every ? ε X ^* is reflexive. We also prove that if ${f : X \rightarrow \mathbb{R} \cup \{+\infty\}}$ is a weakly lower semicontinuous function on the Banach space X and if for every continuous linear functional ? on X the set where the function f ? ? attains its minimum is convex and non-empty then f is convex.     

Young measures generated by sequences in Morrey spaces

Let ${\Omega\subset\mathbb{R}^n}$ be open and bounded. For 1 ≤ p < ∞ and 0 ≤ λ < n, we give a characterization of Young measures generated by sequences of functions ${\{{\bf f}_j\}_{j=1}^\infty}$ uniformly bounded in the Morrey space ${L^{p,\lambda}(\Omega;\mathbb{R}^N)}$ with ${\{\left|{{\bf f}_j}\right|^p\}_{j=1}^\infty}$ equiintegrable. We then treat the case that each f _j = ? u _j for some ${{\bf u}_j\in W^{1,p}(\Omega;\mathbb{R}^N)}$ . As an application of our results, we consider the functional $${\bf u} \mapsto \int\limits_{\Omega}f({\bf x}, {\bf u}({\bf x}), {\bf {\nabla}}{\bf u}({\bf x})){\rm d}{\bf x},$$ and provide conditions that guarantee the existence of a minimizing sequence with gradients uniformly bounded in ${L^{p,\lambda}(\Omega;\mathbb{R}^{N\times n})}$ .     

Value-sets of polynomials at p-adic integers

In this paper we are concerned with the classification of the subsets A of ${\mathbb{Z}_p}$ which occur as images ${f(\mathbb{Z}_p^r)}$ of polynomial functions ${f:\mathbb{Z}_p^r\to \mathbb{Z}_p}$ , limiting ourselves to compact-open subsets (i.e. finite unions of open balls). We shall prove three main results: (i) Every compact-open ${A\subset \mathbb{Z}_p}$ is of the shape ${A=f(\mathbb{Z}_p^r)}$ for suitable r and ${f\in\mathbb{Z}_p[X_1,\ldots ,X_r]}$ . (ii) For each r ₀ there is a compact-open A such that in (i) we cannot take r < r ₀. (iii) For any compact-open set ${A\subset \mathbb{Z}_p}$ there exists a polynomial ${f\in\mathbb{Q}_p[X]}$ such that ${f(\mathbb{Z}_p)=A}$ . We shall also discuss in more detail which sets A can be represented as ${f(\mathbb{Z}_p)}$ for a polynomial ${f\in\mathbb{Z}_p[X]}$ in a single variable.     

Regularity for non-autonomous functionals with almost linear growth

We consider non-autonomous functionals ${\mathcal{F}(u; \Omega)=\int_{\Omega}f(x, Du)\ dx}$ , where the density ${f:\Omega\times\mathbb{R}^{nN}\rightarrow\mathbb{R}}$ has almost linear growth, i.e., $$f(x,\xi)\approx |\xi|\log(1+|\xi|).$$ We prove partial C ^1,?? -regularity for minimizers ${u:\mathbb{R}^n\supset\Omega\rightarrow \mathbb{R}^N}$ under the assumption that D _?? f (x, ??) is H?lder continuous with respect to the x-variable. If the x-dependence is C ¹ we can improve this to full regularity provided additional structure conditions are satisfied.     

A Characterization of Sobolev Spaces on the Sphere and an Extension of Stolarsky’s Invariance Principle to Arbitrary Smoothness

In this paper, we study reproducing kernel Hilbert spaces of arbitrary smoothness on the sphere $\mathbb{S}^{d} \subset\mathbb{R}^{d+1}$ . The reproducing kernel is given by an integral representation using the truncated power function $(\mathbf{x} \cdot\mathbf{z} - t)_{+}^{\beta-1}$ supported on spherical caps centered at z of height t, which reduces to an integral over indicator functions of open spherical caps if β=1, as studied in Brauchart and Dick (Proc. Am. Math. Soc. 141(6):2085–2096, 2013). This is analogous to a generalization of the reproducing kernel to arbitrary smoothness on the unit cube by Temlyakov (J. Complex. 19(3):352–391, 2003). We show that the reproducing kernel is a sum of the Euclidean distance ∥x?y∥ of the arguments of the kernel raised to the power of 2β?1 and an adjustment in the form of a Kampé de Fériet function that ensures positivity of the kernel if 2β?1 is not an even integer; otherwise, a limit process introduces logarithmic terms in the distance. For $\beta\in\mathbb{N}$ , the Kampé de Fériet function reduces to a polynomial, giving a simple closed form expression for the reproducing kernel. Stolarsky’s invariance principle states that the sum of all mutual distances among N points plus a certain multiple of the spherical cap $\mathbb{L}_{2}$ -discrepancy of these points remains constant regardless of the choice of the points. Rearranged differently, it provides a reinterpretation of the spherical cap $\mathbb{L}_{2}$ -discrepancy as the worst-case error of equal-weight numerical integration rules in the Sobolev space over $\mathbb{S}^{d}$ of smoothness (d+1)/2 provided with the reproducing kernel 1?C _d ∥x?y∥ for some constant C _d . Using the new function spaces, we establish an invariance principle for a generalized discrepancy extending the spherical cap $\mathbb{L}_{2}$ -discrepancy and give a reinterpretation as the worst-case error in the Sobolev space over $\mathbb{S}^{d}$ of arbitrary smoothness s=β?1/2+d/2. Previously, Warnock’s formula, which is the analog to Stolarsky’s invariance principle for the unit cube [0,1] ^s , has been generalized using similar techniques in Dick (Ann. Mat. Pura Appl. (4) 187(3):385–403, 2008).     

Metrization of Weighted Graphs

We find a set of necessary and sufficient conditions under which the weight ${w: E \rightarrow \mathbb{R}^{+}}$ on the graph G = (V, E) can be extended to a pseudometric ${d : V \times V \rightarrow \mathbb{R}^{+}}$ . We describe the structure of graphs G for which the set ${\mathfrak{M}_{w}}$ of all such extensions contains a metric whenever w is strictly positive. Ordering ${\mathfrak{M}_{w}}$ by the pointwise order, we have found that the posets $({\mathfrak{M}_{w}, \leqslant)}$ contain the least elements ρ _0,w if and only if G is a complete k-partite graph with ${k \, \geqslant \, 2}$ . In this case the symmetric functions ${f : V \times V \rightarrow \mathbb{R}^{+}}$ , lying between ρ _0,w and the shortest-path pseudometric, belong to ${\mathfrak{M}_{w}}$ for every metrizable w if and only if the cardinality of all parts in the partition of V is at most two.     

On the saturation of subfields of invariants of finite groups

Every subfield $ \mathbb{K} $ (φ) of the field of rational fractions $ \mathbb{K} $ (x ₁,..., x _n ) is contained in a unique maximal subfield of the form $ \mathbb{K} $ (ω). The element ω is said to be generating for the element φ. A subfield of $ \mathbb{K} $ (x ₁,..., x _n ) is said to be saturated if, together with every its element, the subfield also contains the generating element. In the paper, the saturation property is studied for the subfields of invariants $ \mathbb{K} $ (x ₁,..., x _n ) ^G of a finite group G of automorphisms of the field $ \mathbb{K} $ (x ₁..., x _n ).