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 \({\Phi(A)\leqslant \Phi(B) (A, B\in {\rm M_{n}}(\mathbb{D}) )}\) .     

Wandering Subspaces of the Bergman Space and the Dirichlet Space Over {{\mathbb{D}^{n}}}

Doubly commuting invariant subspaces of the Bergman space and the Dirichlet space over the unit polydisc \({\mathbb{D}^n}\) (with \({n \geq 2}\) ) are investigated. We show that for any non-empty subset \({\alpha=\{\alpha_1,\ldots,\alpha_k\}}\) of \({\{1,\ldots,n\}}\) and doubly commuting invariant subspace \({\mathcal{S}}\) of the Bergman space or the Dirichlet space over \({\mathbb{D}^n}\) , restriction of the multiplication operator tuple on \({\mathcal{S}, M_{\alpha}|_\mathcal{S}:=(M_{z_{\alpha_1}}|_\mathcal{S},\ldots, M_{z_{\alpha_k}}|_\mathcal{S})}\) , always possesses generating wandering subspace of the form $$\bigcap_{i=1}^k(\mathcal{S}\ominus z_{\alpha_i}\mathcal{S})$$ .     

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.     

Spectra of tensor triangulated categories over category algebras

Let \({\mathcal{C}}\) be a finite EI category and k be a field. We consider the category algebra \({k\mathcal{C}}\) . Suppose \({\sf{K}(\mathcal{C})=\sf{D}^b(k \mathcal{C}-\sf{mod})}\) is the bounded derived category of finitely generated left modules. This is a tensor triangulated category, and we compute its spectrum in the sense of Balmer. When \({\mathcal{C}=G \propto \mathcal{P}}\) is a finite transporter category, the category algebra becomes Gorenstein, so we can define the stable module category \({\underline{\sf{CM}} k(G \propto \mathcal{P})}\) , of maximal Cohen–Macaulay modules, as a quotient category of \({{\sf{K}}(G \propto \mathcal{P})}\) . Since \({\underline{\sf{CM}} k(G\propto\mathcal{P})}\) is also tensor triangulated, we compute its spectrum as well. These spectra are used to classify tensor ideal thick subcategories of the corresponding tensor triangulated categories.     

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 \({\mathcal{D}}\) .     

A Jost–Pais-Type Reduction of Fredholm Determinants and Some Applications

We study the analog of semi-separable integral kernels in \({\mathcal {H}}\) of the type $$ K(x, x') = \left\{\begin{array}{ll} F_1(x) G_1(x'), \quad& a < x' < x < b,\\ F_2 (x)G_2(x'), \quad& a < x < x' < b,\end{array}\right.$$ where \({-\infty \leqslant a < b \leqslant \infty}\) , and for a.e. \({x \in (a, b)}\) , \({F_j (x) \in \mathcal{B}_2(\mathcal{H}_j, \mathcal{H})}\) and \({G_j(x) \in \mathcal {B}_2(\mathcal {H},\mathcal {H}_j)}\) such that F _j (·) and G _j (·) are uniformly measurable, and $$\begin{array}{ll} || F_j ( \cdot) ||_{\mathcal {B}_2(\mathcal {H}_j,\mathcal {H})} \in L^2((a, b)), ||G_j (\cdot)||_{\mathcal {B}_2(\mathcal {H},\mathcal {H}_j)} \in L^2((a, b)), \quad j=1,2, \end{array}$$ with \({\mathcal {H}}\) and \({\mathcal {H}_j}\) , j = 1, 2, complex, separable Hilbert spaces. Assuming that K(·, ·) generates a trace class operator K in \({L^2((a, b);\mathcal {H})}\) , we derive the analog of the Jost–Pais reduction theory that succeeds in proving that the Fredholm determinant \({{\rm det}_{L^2((a,b);\mathcal{H})}}\) (I ? α K), \({\alpha \in \mathbb{C}}\) , naturally reduces to appropriate Fredholm determinants in the Hilbert spaces \({\mathcal{H}}\) (and \({\mathcal{H}_1 \oplus \mathcal{H}_2}\) ). Explicit applications of this reduction theory to Schrödinger operators with suitable bounded operator-valued potentials are made. In addition, we provide an alternative approach to a fundamental trace formula first established by Pushnitski which leads to a Fredholm index computation of a certain model operator.     

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.     

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.     

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.     

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 \({f, g, h : \mathbb{R}^{n} \to \mathbb{C}}\) .     

A Geometric Uncertainty Principle with an Application to Pleijel’s Estimate

Let \({\Omega \subset \mathbb{R}^2}\) be an open, bounded domain and \({\Omega = \bigcup_{i = 1}^{N} \Omega_{i}}\) be a partition. Denote the Fraenkel asymmetry by \({0 \leq \mathcal{A}(\Omega_i) \leq 2}\) and write $$D(\Omega_i) := \frac{|\Omega_{i}| - {\rm min}_{1 \leq j \leq N}{|\Omega_{j}|}}{|\Omega_{i}|}$$ with \({0 \leq D(\Omega_{i}) \leq 1}\) . For N sufficiently large depending only on \({\Omega}\) , there is an uncertainty principle $$\left(\sum_{i=1}^{N}{\frac{|\Omega_{i}|}{|\Omega|}{\mathcal{A}}(\Omega_i)}\right) + \left(\sum_{i=1}^{N}{\frac{|\Omega_i|}{|\Omega|}D(\Omega_i)}\right) \geq \frac{1}{60000}.$$ The statement remains true in dimensions \({n \geq 3}\) for some constant \({c_{n} > 0}\) . As an application, we give an (unspecified) improvement of Pleijel’s estimate on the number of nodal domains of a Laplacian eigenfunction and an improved inequality for a spectral partition problem.     

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.     

Behaviour at infinity of solutions of some linear functional equations in normed spaces

Let \({\mathbb{K} \in \{\mathbb{R}, \mathbb{C}\}, I = (d, \infty), \phi : I \to I}\) be unbounded continuous and increasing, X be a normed space over \({\mathbb{K}, \mathcal{F} : = \{f \in X^I : {\rm lim}_{t \to \infty} f(t) {\rm exists} \, {\rm in} X\},\hat{a} \in \mathbb{K}, \mathcal{A}(\hat{a}) : = \{\alpha \in \mathbb{K}^I : {\rm lim}_{t \to \infty} \alpha(t) = \hat{a}\},}\) and \({\mathcal{X} : = \{x \in X^I : {\rm lim} \, {\rm sup}_{t \to \infty} \|x(t)\| < \infty\}}\) . We prove that the limit lim _{t → ∞} x(t) exists for every \({f \in \mathcal{F}, \alpha \in \mathcal{A}(\hat{a})}\) and every solution \({x \in \mathcal{X}}\) of the functional equation $$x(\phi(t)) = \alpha(t) x(t) + f(t)$$ if and only if \({|\hat{a}| \neq 1}\) . Using this result we study behaviour of bounded at infinity solutions of the functional equation $$x(\phi^{[k]}(t)) = \sum_{j=0}^{k-1} \alpha_j(t) x (\phi^{[j]}(t)) + f(t),$$ under some conditions posed on functions \({\alpha_j(t), j = 0, 1,\ldots, k - 1,\phi}\) and f.     

On the Relaxed Colouring Game and the Unilateral Colouring Game

In the paper we introduce the new game—the unilateral \({\mathcal{P}}\) -colouring game which can be used as a tool to study the r-colouring game and the (r, d)-relaxed colouring game. Let be given a graph G, an additive hereditary property \({\mathcal {P}}\) and a set C of r colours. In the unilateral \({\mathcal {P}}\) -colouring game similarly as in the r-colouring game, two players, Alice and Bob, colour the uncoloured vertices of the graph G, but in the unilateral \({\mathcal {P}}\) -colouring game Bob is more powerful than Alice. Alice starts the game, the players play alternately, but Bob can miss his move. Bob can colour the vertex with an arbitrary colour from C, while Alice must colour the vertex with a colour from C in such a way that she cannot create a monochromatic minimal forbidden subgraph for the property \({\mathcal {P}}\) . If after |V(G)| moves the graph G is coloured, then Alice wins the game, otherwise Bob wins. The \({\mathcal {P}}\) -unilateral game chromatic number, denoted by \({\chi_{ug}^\mathcal {P}(G)}\) , is the least number r for which Alice has a winning strategy for the unilateral \({\mathcal {P}}\) -colouring game with r colours on G. We prove that the \({\mathcal {P}}\) -unilateral game chromatic number is monotone and is the upper bound for the game chromatic number and the relaxed game chromatic number. We give the winning strategy for Alice to play the unilateral \({\mathcal {P}}\) -colouring game. Moreover, for k ≥  2 we define a class of graphs \({\mathcal {H}_k =\{G|{\rm every \;block \;of\;}G \; {\rm has \;at \;most}\; k \;{\rm vertices}\}}\) . The class \({\mathcal {H}_k }\) contains, e.g., forests, Husimi trees, line graphs of forests, cactus graphs. Let \({\mathcal {S}_d}\) be the class of graphs with maximum degree at most d. We find the upper bound for the \({\mathcal {S}_2}\) -unilateral game chromatic number for graphs from \({\mathcal {H}_3}\) and we study the \({\mathcal {S}_d}\) -unilateral game chromatic number for graphs from \({\mathcal {H}_4}\) for \({d \in \{2,3\}}\) . As the conclusion from these results we obtain the result for the d-relaxed game chromatic number: if \({G \in \mathcal {H}_k}\) , then \({\chi_g^{(d)}(G) \leq k + 2-d}\) , for \({k \in \{3, 4\}}\) and \({d \in \{0, \ldots, k-1\}}\) . This generalizes a known result for trees.     

Representations of algebras in varieties generated by infinite primal algebras

If \({\mathcal{A}}\) is an infinite primal algebra, then we shall represent any algebra in the variety \({V\,(\mathcal{A}}\) ) generated by \({\mathcal{A}}\) as a limit reduced power of \({\mathcal{A}}\) . Furthermore, we show that any homomorphism between algebras in \({V\,(\mathcal{A}}\) ) can be induced by mappings between underlying sets of the limit reduced powers. With this representation of the morphisms between algebras in \({V\,(\mathcal{A}}\) ) at hand, we will construct a category equivalent to the category \({V\,(\mathcal{A}}\) ).     

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 \({L= -\Delta - p | u_{p,\mathcal{K}} | ^{p-1}}\) .     

\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.