A Remark on The Geometry of Spaces of Functions with Prime Frequencies

For any positive integer r, denote by \({\mathcal{P}_{r}}\) the set of all integers \({\gamma \in \mathbb{Z}}\) having at most r prime divisors. We show that \({C_{\mathcal{P}_{r}}(\mathbb{T})}\) , the space of all continuous functions on the circle \({\mathbb{T}}\) whose Fourier spectrum lies in \({\mathcal{P}_{r}}\) , contains a complemented copy of \({\ell^{1}}\) . In particular, \({C_{\mathcal{P}_{r}}(\mathbb{T})}\) is not isomorphic to \({C(\mathbb{T})}\) , nor to the disc algebra \({A(\mathbb{D})}\) . A similar result holds in the L ¹ setting.     

Comparison of fine structural mice via coarse iteration

Let \({\mathcal{M}}\) be a fine structural mouse. Let \({\mathbb{D}}\) be a fully backgrounded \({L[\mathbb{E}]}\) -construction computed inside an iterable coarse premouse S. We describe a process comparing \({\mathcal{M}}\) with \({\mathbb{D}}\) , through forming iteration trees on \({\mathcal{M}}\) and on S. We then prove that this process succeeds.     

Stability of a conditional Cauchy equation on a set of measure zero

In this paper, we prove the Hyers–Ulam stability theorem when \({f, g, h : \mathbb{R} \to \mathbb{R}}\) satisfy $$|f(x + y) - g(x) - h(y)| \leq \epsilon$$ in a set \({\Gamma \subset \mathbb{R}^{2}}\) of measure \({m(\Gamma) = 0}\) , which refines a previous result in Chung (Aequat Math 83:313–320, 2012) and gives an affirmative answer to the question in the paper. As a direct consequence we obtain that if \({f, g, h : \mathbb{R} \to \mathbb{R}}\) satisfy the Pexider equation $$f(x + y) - g(x) - h(y) = 0$$ in \({\Gamma}\) , then the equation holds for all \({x, y \in \mathbb{R}}\) . Using our method of construction of the set, we can find a set \({\Gamma \subset \mathbb{R}^{2n}}\) of 2n-dimensional measure 0 and obtain the above result for the functions \({f, g, h : \mathbb{R}^{n} \to \mathbb{C}}\) .     

Topological genericity of nowhere differentiable functions in the disc algebra

In this paper we introduce a class of functions contained in the disc algebra \({\mathcal{A}(D)}\) . We study functions \({f \in \mathcal{A}(D)}\) which have the property that the continuous periodic function \({u = {\rm Re}f|_{\mathbb{T}}}\) , where \({\mathbb{T}}\) is the unit circle, is nowhere differentiable. We prove that this class is non-empty and instead, generically, every function \({f \in \mathcal{A}(D)}\) has the above property. Afterwards, we strengthen this result by proving that, generically, for every function \({f \in \mathcal{A}(D)}\) , both continuous periodic functions \({u = {\rm Re}f|_\mathbb{T}}\) and \({\tilde{u} = {\rm Im}f|_\mathbb{T}}\) are nowhere differentiable. We avoid any use of the Weierstrass function and we mainly use Baire’s Category Theorem.     

Automorphisms of {{\rm M_{n}}(\mathbb{D})}, Partially Ordered by Rank Subtractivity Ordering

Let \({\mathbb{D}}\) be an arbitrary division ring and \({{\rm M_{n}}(\mathbb{D})}\) be the set of all n × n matrices over \({\mathbb{D}}\) . We define the rank subtractivity or minus partial order on \({{\rm M_{n}}(\mathbb{D})}\) as defined on \({{\rm M_{n}}(\mathbb{C})}\) , i.e., \({A \leqslant B}\) iff rank(B) = rank(A) + rank(B?A). We describe the structure of maps Φ on \({{\rm M_{n}}(\mathbb{D})}\) such that \({A\leqslant B}\) iff \({\Phi(A)\leqslant \Phi(B) (A, B\in {\rm M_{n}}(\mathbb{D}) )}\) .     

Special values of anticyclotomic L-functions modulo λ

The purpose of this article is to generalize some results of Vatsal on the special values of Rankin–Selberg L-functions in an anticyclotomic \({\mathbb{Z}_{p}}\) -extension. Let g be a cuspidal Hilbert modular newform of parallel weight \({(2,\ldots,2)}\) and level \({\mathcal{N}}\) over a totally real field F, and let K/F be a totally imaginary quadratic extension of relative discriminant \({\mathcal{D}}\) . We study the l-adic valuation of the special values \({L(g,\chi,\frac{1}{2})}\) as \({\chi}\) varies over the ring class characters of K of \({\mathcal{P}}\) -power conductor, for some fixed prime ideal \({\mathcal{P}}\) . We prove our results under the only assumption that the prime to \({\mathcal{P}}\) part of \({\mathcal{N}}\) is relatively prime to \({\mathcal{D}}\) .     

Number theoretic applications of a class of Cantor series fractal functions. I

Suppose that \({{(P, Q) \in {\mathbb{N}_{2}^\mathbb{N}} \times {\mathbb{N}_{2}^\mathbb{N}}}}\) and x = E ₀.E ₁ E ₂ · · · is the P-Cantor series expansion of \({x \in \mathbb{R}}\) . We define $$\psi_{P,Q}(x) := {\sum_{n=1}^{\infty}} \frac{{\rm min}(E_n, q_{n}-1)}{q_1 \cdots q_n}.$$ The functions \({\psi_{P,Q}}\) are used to construct many pathological examples of normal numbers. These constructions are used to give the complete containment relation between the sets of Q-normal, Q-ratio normal, and Q-distribution normal numbers and their pairwise intersections for fully divergent Q that are infinite in limit. We analyze the Hölder continuity of \({\psi_{P,Q}}\) restricted to some judiciously chosen fractals. This allows us to compute the Hausdorff dimension of some sets of numbers defined through restrictions on their Cantor series expansions. In particular, the main theorem of a paper by Y. Wang et al. [29] is improved. Properties of the functions \({\psi_{P,Q}}\) are also analyzed. Multifractal analysis is given for a large class of these functions and continuity is fully characterized. We also study the behavior of \({\psi_{P,Q}}\) on both rational and irrational points, monotonicity, and bounded variation. For different classes of ergodic shift invariant Borel probability measures \({\mu_1}\) and \({\mu_2}\) on \({{\mathbb{N}_2^\mathbb{N}}}\) , we study which of these properties \({\psi_{P,Q}}\) satisfies for \({\mu_1 \times \mu_2}\) -almost every (P,Q) \({{\in {\mathbb{N}_{2}^{\mathbb{N}}} \times {\mathbb{N}_{2}^{\mathbb{N}}}}}\) . Related classes of random fractals are also studied.     

Badly approximable points on planar curves and a problem of Davenport

Let \({\mathcal {C}}\) be two times continuously differentiable curve in \({\mathbb {R}}^2\) with at least one point at which the curvature is non-zero. For any \(i,j \geqslant 0\) with \(i+j =1\) , let \({\mathbf {Bad}}(i,j)\) denote the set of points \((x,y) \in {\mathbb {R}}^2\) for which \( \max \{ \Vert qx\Vert ^{1/i}, \, \Vert qy\Vert ^{1/j} \} > c/q \) for all \( q \in {\mathbb {N}}\) . Here \(c = c(x,y)\) is a positive constant. Our main result implies that any finite intersection of such sets with \({\mathcal {C}}\) has full Hausdorff dimension. This provides a solution to a problem of Davenport dating back to the sixties.     

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.     

The {\mathcal {A}}-Truncated K-Moment Problem

Let \({\mathcal {A}}\subseteq {\mathbb {N}}^n\) be a finite set, and \(K\subseteq {\mathbb {R}}^n\) be a compact semialgebraic set. An \({\mathcal {A}}\) -truncated multisequence ( \({\mathcal {A}}\) -tms) is a vector \(y=(y_{\alpha })\) indexed by elements in \({\mathcal {A}}\) . The \({\mathcal {A}}\) -truncated \(K\) -moment problem ( \({\mathcal {A}}\) -TKMP) concerns whether or not a given \({\mathcal {A}}\) -tms \(y\) admits a \(K\) -measure \(\mu \) , i.e., \(\mu \) is a nonnegative Borel measure supported in \(K\) such that \(y_\alpha = \int _K x^\alpha \mathtt {d}\mu \) for all \(\alpha \in {\mathcal {A}}\) . This paper proposes a numerical algorithm for solving \({\mathcal {A}}\) -TKMPs. It aims at finding a flat extension of \(y\) by solving a hierarchy of semidefinite relaxations \(\{(\mathtt {SDR})_k\}_{k=1}^\infty \) for a moment optimization problem, whose objective \(R\) is generated in a certain randomized way. If \(y\) admits no \(K\) -measures and \({\mathbb {R}}[x]_{{\mathcal {A}}}\) is \(K\) -full (there exists \(p \in {\mathbb {R}}[x]_{{\mathcal {A}}}\) that is positive on \(K\) ), then \((\mathtt {SDR})_k\) is infeasible for all \(k\) big enough, which gives a certificate for the nonexistence of representing measures. If \(y\) admits a \(K\) -measure, then for almost all generated \(R\) , this algorithm has the following properties: i) we can asymptotically get a flat extension of \(y\) by solving the hierarchy \(\{(\mathtt {SDR})_k\}_{k=1}^\infty \) ; ii) under a general condition that is almost sufficient and necessary, we can get a flat extension of \(y\) by solving \((\mathtt {SDR})_k\) for some \(k\) ; iii) the obtained flat extensions admit a \(r\) -atomic \(K\) -measure with \(r\le |{\mathcal {A}}|\) . The decomposition problems for completely positive matrices and sums of even powers of real linear forms, and the standard truncated \(K\) -moment problems, are special cases of \({\mathcal {A}}\) -TKMPs. They can be solved numerically by this algorithm.     

\mathcal {A}_{p, {\mathbb {E}}} Weights,Maximal Operators,and Hardy Spaces Associated with a Family of General Sets

Suppose that \({\mathbb {E}}:=\{E_r(x)\}_{r\in {\mathcal {I}}, x\in X}\) is a family of open subsets of a topological space \(X\) endowed with a nonnegative Borel measure \(\mu \) satisfying certain basic conditions. We establish an \(\mathcal {A}_{{\mathbb {E}}, p}\) weights theory with respect to \({\mathbb {E}}\) and get the characterization of weighted weak type (1,1) and strong type \((p,p)\) , \(1<p\le \infty \) , for the maximal operator \({\mathcal {M}}_{{\mathbb {E}}}\) associated with \({\mathbb {E}}\) . As applications, we introduce the weighted atomic Hardy space \(H^1_{{\mathbb {E}}, w}\) and its dual \(BMO_{{\mathbb {E}},w}\) , and give a maximal function characterization of \(H^1_{{\mathbb {E}},w}\) . Our results generalize several well-known results.     

Pro-\mathcal {C} completions of orientable \text{ PD }^3-pairs

Let \({\mathcal {C}}\) be a class of finite groups. We study some sufficient conditions for the pro- \({\mathcal {C}}\) completion of an orientable \(\text{ PD }^3\) -pair over \(\mathbb {Z}\) to be an orientable profinite \(\text{ PD }^3\) -pair over \(\mathbb {F}_p\) . More results are proven for the pro- \(p\) completion of \(\text{ PD }^3\) -pairs.     

The Cubic Complex Moment Problem

Let \({s = \{s_{jk}\}_{0 \leq j+k \leq 3}}\) be a given complex-valued sequence. The cubic complex moment problem involves determining necessary and sufficient conditions for the existence of a positive Borel measure \({\sigma}\) on \({\mathbb{C}}\) (called a representing measure for s) such that \({s_{jk} = \int_{\mathbb{C}}\bar{z}^j z^k d\sigma(z)}\) for \({0 \leq j + k \leq 3}\) . Put $$\Phi = \left(\begin{array}{lll} s_{00} & s_{01} & s_{10} \\s_{10} & s_{11} & s_{20} \\s_{01} & s_{02} & s_{11}\end{array}\right), \quad \Phi_z = \left(\begin{array}{lll}s_{01} & s_{02} & s_{11} \\s_{10} & s_{12} & s_{21} \\s_{02} & s_{03} & s_{12}\end{array} \right)\quad {\rm and}\quad\Phi_{\bar{z}} = (\Phi_z)^*.$$ If \({\Phi \succ 0}\) , then the commutativity of \({\Phi^{-1} \Phi_z}\) and \({\Phi^{-1} \Phi_{\bar{z}}}\) is necessary and sufficient for the existence a 3-atomic representing measure for s. If \({\Phi^{-1} \Phi_z}\) and \({\Phi^{-1} \Phi_{\bar{z}}}\) do not commute, then we show that s has a 4-atomic representing measure. The proof is constructive in nature and yields a concrete parametrization of all 4-atomic representing measures of s. Consequently, given a set \({K \subseteq \mathbb{C}}\) necessary and sufficient conditions are obtained for s to have a 4-atomic representing measure \({\sigma}\) which satisfies \({{\rm supp} \sigma \cap K \neq \emptyset}\) or \({{\rm supp} \sigma \subseteq K}\) . The cases when \({K = \overline{\mathbb{D}}}\) and \({K = \mathbb{T}}\) are considered in detail.     

Forbidden Triples Containing a Complete Graph and a Complete Bipartite Graph of Small Order

For a graph G and a set \({\mathcal{F}}\) of connected graphs, G is said be \({\mathcal{F}}\) -free if G does not contain any member of \({\mathcal{F}}\) as an induced subgraph. We let \({\mathcal{G} _{3}(\mathcal{F})}\) denote the set of all 3-connected \({\mathcal{F}}\) -free graphs. This paper is concerned with sets \({\mathcal{F}}\) of connected graphs such that \({\mathcal{F}}\) contains no star, \({|\mathcal{F}|=3}\) and \({\mathcal{G}_{3}(\mathcal{F})}\) is finite. Among other results, we show that for a connected graph T( ≠ K ₁) which is not a star, \({\mathcal{G}_{3}(\{K_{4},K_{2,2},T\})}\) is finite if and only if T is a path of order at most 6.     

Reeb orbits trapped by Denjoy minimal sets

Let φ be any flow on T ⁿ obtained as the suspension of a smooth diffeomorphism of \({T^{n-1}}\) , and let \({\mathcal {A}}\) be any compact invariant set of φ. We realize \({(\mathcal{A}, \varphi|_{\mathcal{A}})}\) up to reparametrization as an invariant set of the Reeb flow of a contact form on \({\mathbb{R}^{2n+1}}\) equal to the standard contact form outside a compact set and defining the standard contact structure on all of \({\mathbb{R}^{2n+1}}\) . This uses the method from Geiges, Röttgen, and Zehmisch.     

Weak theories of concatenation and minimal essentially undecidable theories

We consider weak theories of concatenation, that is, theories for strings or texts. We prove that the theory of concatenation \({\mathsf{WTC}^{-\varepsilon}}\) , which is a weak subtheory of Grzegorczyk’s theory \({\mathsf{TC}^{-\varepsilon}}\) , is a minimal essentially undecidable theory, that is, the theory \({\mathsf{WTC}^{-\varepsilon}}\) is essentially undecidable and if one omits an axiom scheme from \({\mathsf{WTC}^{-\varepsilon}}\) , then the resulting theory is no longer essentially undecidable. Moreover, we give a positive answer to Grzegorczyk and Zdanowski’s conjecture that ‘The theory \({\mathsf{TC}^{-\varepsilon}}\) is a minimal essentially undecidable theory’. For the alternative theories \({\mathsf{WTC}}\) and \({\mathsf{TC}}\) which have the empty string, we also prove that the each theory without the neutrality of \({\varepsilon}\) is to be such a theory too.     

Anderson’s Orthogonality Catastrophe for One-Dimensional Systems

The overlap, \({\mathcal{D}_N}\) , between the ground state of N free fermions and the ground state of N fermions in an external potential in one spatial dimension is given by a generalized Gram determinant. An upper bound is \({\mathcal{D}_N\leq\exp(-\mathcal{I}_N)}\) with the so-called Anderson integral \({\mathcal{I}_N}\) . We prove, provided the external potential satisfies some conditions, that in the thermodynamic limit \({\mathcal{I}_N = \gamma\ln N + O(1)}\) as \({N\to\infty}\) . The coefficient γ > 0 is given in terms of the transmission coefficient of the one-particle scattering matrix. We obtain a similar lower bound on \({\mathcal{D}_N}\) concluding that \({\tilde{C} N^{-\tilde{\gamma}} \leq \mathcal{D}_N \leq CN^{-\gamma}}\) with constants C, \({\tilde{C}}\) , and \({\tilde{\gamma}}\) . In particular, \({\mathcal{D}_N\to 0}\) as \({N\to\infty}\) which is known as Anderson’s orthogonality catastrophe.     

Sign-changing stationary solutions and blowup for a nonlinear heat equation in dimension two

Consider the nonlinear heat equation $$v_t -\Delta v=|v|^{p-1}v \qquad \qquad \qquad (NLH)$$ in the unit ball of \({\mathbb{R}^2}\) , with Dirichlet boundary condition. Let \({u_{p,\mathcal{K}}}\) be a radially symmetric, sign-changing stationary solution having a fixed number \({\mathcal{K}}\) of nodal regions. We prove that the solution of (NLH) with initial value \({\lambda u_{p,\mathcal{K}}}\) blows up in finite time if |λ ?1| > 0 is sufficiently small and if p is sufficiently large. The proof is based on the analysis of the asymptotic behavior of \({u_{p,\mathcal{K}}}\) and of the linearized operator \({L= -\Delta - p | u_{p,\mathcal{K}} | ^{p-1}}\) .