In this paper, we consider projective deformation of the geodesic system of Finsler spaces by holonomy invariant functions. Starting with a Finsler spray $S$ and a holonomy invariant function ${\mathcal{P}}$, we investigate the metrizability property of the projective deformation $\widetilde{S}=S-2\unicode[STIX]{x1D706}{\mathcal{P}}{\mathcal{C}}$. We prove that for any holonomy invariant nontrivial function ${\mathcal{P}}$ and for almost every value $\unicode[STIX]{x1D706}\in \mathbb{R}$, such deformation is not Finsler metrizable. We identify the cases where such deformation can lead to a metrizable spray. In these cases, the holonomy invariant function ${\mathcal{P}}$ is necessarily one of the principal curvatures of the geodesic structure.

We determine the integral Chow and cohomology rings of the moduli stack $\mathcal {B}_{r,d}$ of rank r, degree d vector bundles on $\mathbb {P}^1$ -bundles. We work over a field k of arbitrary characteristic. We first show that the rational Chow ring $A_{\mathbb {Q}}^*(\mathcal {B}_{r,d})$ is a free $\mathbb {Q}$ -algebra on $2r+1$ generators. The isomorphism class of this ring happens to be independent of d. Then, we prove that the integral Chow ring $A^*(\mathcal {B}_{r,d})$ is torsion-free and provide multiplicative generators for $A^*(\mathcal {B}_{r,d})$ as a subring of $A_{\mathbb {Q}}^*(\mathcal {B}_{r,d})$ . From this description, we see that $A^*(\mathcal {B}_{r,d})$ is not finitely generated as a $\mathbb {Z}$ -algebra. Finally, when $k = \mathbb {C}$ , the cohomology ring of $\mathcal {B}_{r,d}$ is isomorphic to its Chow ring.

We show that geodesics in $\mathbf {H}$ attached to a maximal split torus or a real quadratic torus in $GL_{2, \mathbf {Q}}$ are the only irreducible algebraic curves in $\mathbf {H}$ whose image in $\mathbf {R}^2$ via the j-invariant is contained in an algebraic curve.

Motivated by Altshuler’s famous characterization of the unit vector basis of $c_0$ or $\ell _p$ among symmetric bases (Altshuler [1976, Israel Journal of Mathematics, 24, 39–44]), we obtain similar characterizations among democratic bases and among bidemocratic bases. We also prove a separate characterization of the unit vector basis of $\ell _1$.

A twisting system is one of the major tools to study graded algebras; however, it is often difficult to construct a (nonalgebraic) twisting system if a graded algebra is given by generators and relations. In this paper, we show that a twisted algebra of a geometric algebra is determined by a certain automorphism of its point variety. As an application, we classify twisted algebras of three-dimensional geometric Artin–Schelter regular algebras up to graded algebra isomorphism.

We discuss the question of extending homeomorphisms between closed subsets of the Cantor cube $D^{\tau }$. It is established that any homeomorphism between two closed negligible subsets of $D^{\tau }$ can be extended to an autohomeomorphism of $D^{\tau }$.

Let $F_{2^n}$ be the Frobenius group of degree $2^n$ and of order $2^n ( 2^n-1)$ with $n \ge 4$ . We show that if $K/\mathbb {Q} $ is a Galois extension whose Galois group is isomorphic to $F_{2^n}$ , then there are $\dfrac {2^{n-1} +(-1)^n }{3}$ intermediate fields of $K/\mathbb {Q} $ of degree $4 (2^n-1)$ such that they are not conjugate over $\mathbb {Q}$ but arithmetically equivalent over $\mathbb {Q}$ . We also give an explicit method to construct these arithmetically equivalent fields.

In this paper, we determine the homotopy types of the Morse complexes of certain collections of simplicial complexes by studying dominating vertices or strong collapses. We show that if K contains two leaves that share a common vertex, then its Morse complex is strongly collapsible and hence has the homotopy type of a point. We also show that the pure Morse complex of a tree is strongly collapsible, thereby recovering as a corollary a result of Ayala et al. (2008, Topology and Its Applications 155, 2084–2089). In addition, we prove that the Morse complex of a disjoint union $K\sqcup L$ is the Morse complex of the join $K*L$ . This result is used to compute the homotopy type of the Morse complex of some families of graphs, including Caterpillar graphs, as well as the automorphism group of a disjoint union for a large collection of disjoint complexes.

We define smooth notions of concordance and sliceness for spatial graphs. We prove that sliceness of a spatial graph is equivalent to a condition on a set of linking numbers together with sliceness of a link associated with the graph. This generalizes the result of Taniyama for $\theta $-curves.

We study the following decision problem: given an exponential equation $a_1g_1^{x_1}a_2g_2^{x_2}\dots a_ng_n^{x_n}=1$ over a recursively presented group G, decide if it has a solution with all $x_i$ in $\mathbb {Z}$. We construct a finitely presented group G where this problem is decidable for equations with one variable and is undecidable for equations with two variables. We also study functions estimating possible solutions of such an equation through the lengths of its coefficients with respect to a given generating set of G. Another result concerns Turing degrees of some natural fragments of the above problem.

Let R be an integral domain with $qf(R)=K$ , and let $F(R)$ be the set of nonzero fractional ideals of R. Call R a dually compact domain (DCD) if, for each $I\in F(R)$ , the ideal $I_{v}=(I^{-1})^{-1}$ is a finite intersection of principal fractional ideals. We characterize DCDs and show that the class of DCDs properly contains various classes of integral domains, such as Noetherian, Mori, and Krull domains. In addition, we show that a Schreier DCD is a greatest common divisor (GCD) (Greatest Common Divisor) domain with the property that, for each $A\in F(R)$ , the ideal $A_{v}$ is principal. We show that a domain R is G-Dedekind (i.e., has the property that $A_{v}$ is invertible for each $A\in F(R)$ ) if and only if R is a DCD satisfying the property $\ast :$ For all pairs of subsets $\{a_{1},\ldots ,a_{m}\},\{b_{1},\ldots ,b_{n}\}\subseteq K\backslash \{0\}, (\cap _{i=1}^{m}(a_{i})(\cap _{j=1}^{n}(b_{j}))=\cap _{i,j=1}^{m,n}(a_{i}b_{j})$ . We discuss what the appropriate names for G-Dedekind domains and related notions should be. We also make some observations about how the DCDs behave under localizations and polynomial ring extensions.

The goal of the paper is to obtain analogs of the sampling theorems and of the Riesz–Boas interpolation formulas which are relevant to the discrete Hilbert and Kak–Hilbert transforms in $l^{2}$ .

It is well known that for any inner function $\theta $ defined in the unit disk $\mathbb {D}$, the following two conditions: (i) there exists a sequence of polynomials $\{p_n\}_n$ such that $\lim _{n \to \infty } \theta (z) p_n(z) = 1$ for all $z \in \mathbb {D}$ and (ii) $\sup _n \| \theta p_n \|_\infty < \infty $, are incompatible, i.e., cannot be satisfied simultaneously. However, it is also known that if we relax the second condition to allow for arbitrarily slow growth of the sequence $\{ \theta (z) p_n(z)\}_n$ as $|z| \to 1$, then condition (i) can be met for some singular inner function. We discuss certain consequences of this fact which are related to the rate of decay of Taylor coefficients and moduli of continuity of functions in model spaces $K_\theta $. In particular, we establish a variant of a result of Khavinson and Dyakonov on nonexistence of functions with certain smoothness properties in $K_\theta $, and we show that the classical Aleksandrov theorem on density of continuous functions in $K_\theta $ is essentially optimal. We consider also the same questions in the context of de Branges–Rovnyak spaces $\mathcal {H}(b)$ and show that the corresponding approximation result also is optimal.

For a three-dimensional quantum polynomial algebra $A=\mathcal {A}(E,\sigma )$ , Artin, Tate, and Van den Bergh showed that A is finite over its center if and only if $|\sigma |<\infty $ . Moreover, Artin showed that if A is finite over its center and $E\neq \mathbb P^{2}$ , then A has a fat point module, which plays an important role in noncommutative algebraic geometry; however, the converse is not true in general. In this paper, we will show that if $E\neq \mathbb P^{2}$ , then A has a fat point module if and only if the quantum projective plane ${\sf Proj}_{\text {nc}} A$ is finite over its center in the sense of this paper if and only if $|\nu ^{*}\sigma ^{3}|<\infty $ where $\nu $ is the Nakayama automorphism of A. In particular, we will show that if the second Hessian of E is zero, then A has no fat point module.

A result of Corfield, Sati, and Schreiber asserts that $\mathfrak {gl}_{n}$-weight systems associated with the defining representation are quantum states. In this short note, we extend this result to all $\mathfrak {gl}_{n}$-weight systems corresponding to labeling by symmetric and exterior powers of the defining representation.

Consider the extended hull of a weak model set together with its natural shift action. Equip the extended hull with the Mirsky measure, which is a certain natural pattern frequency measure. It is known that the extended hull is a measure-theoretic factor of some group rotation, which is called the underlying torus. Among other results, in the article Periods and factors of weak model sets, we showed that the extended hull is isomorphic to a factor group of the torus, where certain periods of the window of the weak model set have been factored out. This was proved for weak model sets having a compact window. In this note, we argue that the same results hold for arbitrary measurable and relatively compact windows. Our arguments crucially rely on Moody’s work on uniform distribution in model sets. We also discuss implications for the diffraction of such weak model sets and discuss a new class of examples which are generic for the Mirsky measure.

Consider three normalized cuspidal eigenforms of weight $2$ and prime level p. Under the assumption that the global root number of the associated triple product L-function is $+1$ , we prove that the complex Abel–Jacobi image of the modified diagonal cycle of Gross–Kudla–Schoen on the triple product of the modular curve $X_0(p)$ is torsion in the corresponding Hecke isotypic component of the Griffiths intermediate Jacobian. The same result holds with the complex Abel–Jacobi map replaced by its étale counterpart. As an application, we deduce torsion properties of Chow–Heegner points associated with modified diagonal cycles on elliptic curves of prime conductor with split multiplicative reduction. The approach also works in the case of composite square-free level.

Hardy kernels are a useful tool to define integral operators on Hilbertian spaces like $L^2(\mathbb R^+)$ or $H^2(\mathbb C^+)$ . These kernels entail an algebraic $L^1$ -structure which is used in this work to study the range spaces of those operators as reproducing kernel Hilbert spaces. We obtain their reproducing kernels, which in the $L^2(\mathbb R^+)$ case turn out to be Hardy kernels as well. In the $H^2(\mathbb C^+)$ scenario, the reproducing kernels are given by holomorphic extensions of Hardy kernels. Other results presented here are theorems of Paley–Wiener type, and a connection with one-sided Hilbert transforms.

We improve upon the traditional error term in the truncated Perron formula for the logarithm of an L-function. All our constants are explicit.

We investigate the weighted $L_p$ affine surface areas which appear in the recently established $L_p$ Steiner formula of the $L_p$ Brunn–Minkowski theory. We show that they are valuations on the set of convex bodies and prove isoperimetric inequalities for them. We show that they are related to f divergences of the cone measures of the convex body and its polar, namely the Kullback–Leibler divergence and the Rényi divergence.