Papers

Leveraging the neural architectures which we introduced in arXiv:2109.13512v4, we show a global universal approximation theorem in the topology of $L^p(\mu)$, where $1\le p<\infty$ and $\mu$ is a Radon probability measure on a suitable infinite dimensional topological space $\mathfrak X$. Namely, any function $f:\mathfrak X\to \mathbb R$ in $L^p(\mu)$ can be approximated to any degree of accuracy by suitable infinite dimensional architectures. These architectures can be in turn approximated by almost classical neural networks which are specified by a finite number of parameters only. The vectorial case (where $f=f(x)\in E$ and $E$ is a Banach space) is also considered and analogous results are obtained.

We introduce the tame factorization property for triples of Fréchet spaces: a triple $(E,G,F)$ has the tame factorization property, denoted by $(E,G,F) \in \mathfrak{TF}$, if there exists a nondecreasing function $S:\mathbb{N}\to\mathbb{N}$ such that every operator from $E$ to $F$ factoring through $G$ is $S$-tame. We give a complete characterization of $\mathfrak{TF}$ in terms of operator seminorm estimates. We specialize this characterization to triples in which one or more of the spaces are Köthe spaces, putting explicit conditions on the Köthe matrices, and we describe how $\mathfrak{TF}$ behaves under projective tensor products of Köthe spaces. Finally, we show that a triple of Köthe spaces has the tame factorization property if and only if the family of operators that factor as products of two quasi-diagonal operators is tame.

A noetherian ring is called G-regular when all finitely generated Gorenstein projective modules are projective. In this paper, we study rings satisfying the stronger condition that all Gorenstein projective modules are projective, which we call strongly G-regular. We show that the notion of strongly G-regular rings is closely related to that of quasi-dominant rings introduced by Takahashi and to the covariant/contravariant finiteness of a certain thick subcategory. We also answer a series of questions due to Chen in the negative, showing that the Gorenstein projective analogue of the Auslander-Ringel-Tachikawa theorem fails even for commutative local artin algebras which are weakly Gorenstein in the sense of Ringel and Zhang.

We show that, up to isomorphism, the number of $Q$-points is either finite, $2^{\mathfrak{d}}$ or $2^{\mathfrak{c}}$. This answers a question asked by Borodulin-Nadzieja, Martínez-Celis, Morawski and Świerczyńska, and by Halbeisen and the authors. We also show that under mild hypotheses, the existence of infinitely many $Q$-points implies the existence of non-atomic $Q$-measures, and of $2^{\mathfrak{c}}$-many Tukey-top $Q$-points, strengthening results of Raghavan and of Borodulin-Nadzieja et al..

We show that isoclasses of iterated tilted algebras in type $A_n$ are in bijection with non-crossing spanning trees up to rotation on a convex n+1-gon. This is done by constructing a relationship between iterated tilted algebras up to isomorphism and exceptional sets up to isomorphic Hom-Ext quiver.

Let $G$ be a finite group. We study the additive map on the Burnside ring of $G$ that sends an orbit $[G/H]$ to $[G/\operatorname{core}_G(H)]$, which we call the normal-core compression, and investigate its behavior on units. For a normal subgroup $N\trianglelefteq G$, we show that the cumulative $N$-core coefficient is equal to the sum of the orbit-basis coefficients of the $N$-fixed-point element over $G/N$. Consequently, for a unit this coefficient takes only the values $-1,0,1$, while the individual core coefficients are recovered by Möbius inversion on the lattice of normal subgroups. Using Yoshida's criterion, we further express the cumulative core coefficient in terms of the $N$-mark and a linear character of $G/N$. This yields a necessary and sufficient condition for the existence of a unit with nonzero $N$-core coefficient in the normal partial Burnside ring: $G/N$ must be an elementary abelian $2$-group. On the other hand, the units mapped to $1_{\Omega(G)}$ by the core compression, called core-trivial units, are characterized by the condition that all marks at core-cyclic subgroups are equal to $1$. Using primitive idempotents of the rational Burnside ring and Yoshida's criterion, we realize the group of core-trivial units as the kernel of an ${\mathbb F}_2$-linear defect map. Finally, we obtain a splitting along normal quotients and a direct-sum decomposition along direct products for this defect map. Combining the direct-product decomposition with an explicit nontrivial example for $S_4$, we show that the ranks of core-trivial unit groups are unbounded among finite groups.

In this article we study the analog of Kakeya needle problem for $(n-1)$-rectifiable sets in $\mathbb{R}^n$ and construct the related Nikodym type sets. The novelty of our approach lies in combining three ingredients: the two-dimensional Venetian blind-type construction for isometries in $\mathbb{R}^n$; the normal geometry of a $(n-1)$-rectifiable set viewed in the projective setting; and measure estimates of moving $(n-1)$-rectifiable sets by isometries in $\mathbb{R}^n$. Together, these ingredients enable us to move $(n-1)$-rectifiable sets along paths of isometries in $\mathbb{R}^n$ and cover a Lebesgue null set.

We determine necessary and sufficient conditions for complete monotonicity of the Srivastava–Tomovski extension, a generalized Mittag–Leffler kernel arising in fractional calculus, while separating the defining generalized-Wright series from its positive-axis realization when the series is not entire. For positive parameters, let \(\Delta=1+\alpha-\kappa\). In the entire regime \(\Delta>0\), \(E_{\alpha,\beta}^{\gamma,\kappa}(-x)\) is completely monotone on \((0,\infty)\) if and only if \(\alpha\leq\kappa\) and \(\kappa\beta\geq\alpha\gamma\). For arbitrary positive parameters, the corresponding positive-axis realization satisfies the same criterion; when \(0<\alpha<\kappa\), it is defined by a second-kind Wright kernel and remains meaningful through the finite-radius and zero-radius phases of the defining series. We also identify the Bernstein measure in every admissible case. In the interior region it is, after normalization, the pushforward under \(t\mapsto t^\kappa\) of a power-biased Wright distribution; at \(\alpha=\kappa\) it becomes a powered beta law for \(\beta>\gamma\) and the atom \(\Gamma(\gamma)^{-1}\delta_1\) for \(\beta=\gamma\). Necessity in the excluded regions follows from a Mellin-transform zero, uniqueness of finite signed Laplace transforms, and moment-support asymptotics. The Prabhakar criterion is recovered by setting \(\kappa=1\).

This paper reviews the subject of equivalence after extension in the context of Toeplitz operators, truncated Toeplitz operators, and further generalizations including dual and multiband truncated Toeplitz operators. Paired operators are shown to play a key role here. Properties under investigation by these methods include the invertibility and Fredholm properties of generalized Toeplitz operators, with new results showing how the kernels of such operators can often be fully described given only limited information.

We prove a sharp volume gap estimate: if an $n$-dimensional compact Kähler manifold $(X, \omega)$ satisfies $\mathrm{Ric}(\omega)\ge (n+1)\omega$ and $X\not\cong \mathbb{P}^n$, then $\mathrm{vol}(X, \omega)\le \frac{2n^n}{(n+1)^n}\mathrm{vol}(\mathbb{P}^n,\omega_{\mathrm{FS}})=\frac{2^{n+1} \, \pi^n \, n^n}{(n+1)^n}$. Moreover $\mathrm{vol}(X, \omega)= \frac{2^{n+1} \, \pi^n \, n^n}{(n+1)^n}$ occurs if and only if $(X, \omega)$ is biholomorphically isometric to the Kähler-Einstein metric on the quadric hypersurface $Q^n$ or on the product $\mathbb{P}^1\times \mathbb{P}^{n-1}$. We also obtain sharp volume gap estimates for K-semistable toric log Fano pairs.

Recent work produced a linearly dependent system of twelve time–frequency shifts of a Schwartz function, disproving the HRT conjecture. We show that four shifts already suffice, and hence that four is the smallest possible cardinality of a dependent finite Gabor system. More precisely, set $\alpha=\frac13+10^{-12}\sqrt2$ and $\beta=\frac13+10^{-12}\sqrt3$. We construct a nonzero complex-valued function $f\in\mathcal S(\mathbb R)$ and $\lambda\ne0$ such that $\left(I+\frac12W(1,0)+\frac12W(0,1/2)\right)W(\alpha,\beta/2)f=\lambda f$, where $W$ denotes the Weyl time–frequency shift. Since every system of at most three shifts of a nonzero $L^2(\mathbb R)$ function is linearly independent, this gives the sharp cardinality threshold. The construction uses the rank-two Zak bundle naturally associated with the covolume-$1/2$ lattice generated by $(1,0)$ and $(0,1/2)$. At the rational translation $(1/3,1/3)$, the three-step return has a uniformly dominated contracting line. A finite outward-rounded interval certificate proves that this line is topologically trivial. A quantitative perturbation argument carries the dominated line to the explicit algebraic translation above. A winding calculation and a Diophantine cohomological equation then flatten its scalar multiplier, and inverse Zak folding produces the required Schwartz function.

The coefficients that determine the Hilbert series of the superspace coinvariant ring are indexed by hook-shaped partitions. We give a manifestly positive combinatorial interpretation of these coefficients, together with several generating functions for them. Specializing this Hilbert series at $u=-q^2$, we show that its coefficients are differences of binomial coefficients. Consequently, this proves a conjecture of Sagan–Swanson (2024) that these coefficients are palindromic up to sign. More generally, for every $m \geq 1$ we obtain closed-form expressions for the $u = -q^m$ specialization.

For each phase $\Theta\in(\pi/2,\pi)$, we construct a smooth, bounded, uniformly convex domain in $\mathbb R^3$ for which the zero-Dirichlet solution of the special Lagrangian equation has a nonconvex negative sublevel set.

Let $B^H$ be a one-dimensional fractional Brownian motion with Hurst index $H\in(1/2,1)$. We study the small-mass limit of the kinetic equation \[ dX_t^\mu=Y_t^\mu\,dt,\qquad \mu\,dY_t^\mu=b(X_t^\mu)\,dt-Y_t^\mu\,dt +\sigma(X_t^\mu)\,dB_t^H, \] where the noise coefficient is state dependent and uniformly nondegenerate. The limiting equation is the Young differential equation \[ dX_t=b(X_t)\,dt+\sigma(X_t)\,dB_t^H. \] Under uniform ellipticity and strict dissipativity of the transformed drift, we prove that, for every $t_0>0$ and every $\rho<2H-1$, \[ \sup_{t\ge t_0}d_{TV}\big(X_t^\mu,X_t\big) \le C_{t_0,\rho}\mu^\rho . \] The estimate is uniform on the entire half-line. We also construct stationary solutions on a two-sided fractional-noise space and obtain convergence of their one-time marginals in total variation. The exponent $2H-1$ is identified as the natural endpoint: it is generated by the quadratic velocity term after the Lamperti transform, and it degenerates as $H\downarrow1/2$. A nonconstant uniformly elliptic example is included.

We propose and analyze a frequency-stable boundary integral equation method for scalar wave scattering by periodic diffraction gratings in two dimensions. The standard quasi-periodic Green function is singular as a function of the frequency at Wood anomaly frequencies, which prevents a direct frequency-uniform error analysis of conventional integral equation discretizations. Our method is based on a modified quasi-periodic Green function obtained by subtracting the singular Rayleigh modes and introducing a finite-dimensional correction that restores the radiation condition. The resulting combined-field boundary integral equation is shown to be uniquely solvable whenever the underlying diffraction problem is uniquely solvable, and the corresponding scattered field depends continuously on the frequency, including at Wood anomalies. We then construct a fully discrete Nyström scheme using an Ewald representation of the Green function and a quadrature rule adapted to the logarithmic singularity of the kernel. For compact frequency intervals that may contain Wood anomaly frequencies, we prove unique solvability of the discrete system for sufficiently large discretization and truncation parameters and establish a frequency-uniform error estimate and bound on the condition number. Numerical examples illustrate the convergence of the scheme and its stability across Wood anomaly frequencies.

Analog error-correcting codes (Analog ECCs), introduced by Roth [1], address errors in vector-matrix multiplication arising from analog noise and sparse outliers in in-memory computing. A fundamental open problem concerns the lower bound on the single-error correction threshold $\Gamma_2(\mathcal C)$ for real $[n,k]$ linear codes with fixed redundancy $r=n-k\geq 2$. Li et al. [2] recently established that for redundancy $r=2$, every real $[n,n-2]$ linear code $\mathcal{C}$ satisfies $\Gamma_2(\mathcal C)\geq \csc^2(\frac{\pi}{2n})$, resolving an open problem in [1], and showed that, for every fixed $r \geq 2$, there exists a class of $[n,k]$ linear code $\mathcal{C}$ over $\mathbb{R}$ such that $\Gamma_2(\mathcal{C}) \leq O(n^{1+\frac{1}{r-1}})$. This paper proves the matching converse in [2]. For every $[n,k]$ linear code $\mathcal{C}\subseteq \mathbb R^n$ with fixed redundancy $2\leq r<n$, we show that \[ \Gamma_2(\mathcal C)\ge \frac{a_r}{\sqrt{r}\,\beta_{r-1}\,2^{\frac{1}{r-1}}}\cdot n^{1+\frac{1}{r-1}}, \] where $a_r =\frac{\Gamma(\frac{r}{2})}{\sqrt{\pi}\,\Gamma(\frac{r+1}{2})}$ and $\beta_d = \left(\frac{d\pi^{d-1}|\mathbb S^d|}{|\mathbb S^{d-1}|}\right)^{\frac{1}{d}}$ for positive integer $d$. Here $\mathbb S^d$ denotes the unit sphere in $\mathbb R^{d+1}$, $|\mathbb S^d|$ its surface area, and $\Gamma(\cdot)$ the Gamma function. In particular, we further show that $\Gamma_2(\mathcal C)\geq \frac{1}{4\pi\sqrt{3}r}\cdot n^{1+\frac{1}{r-1}}$. Together with the upper bound in [2], this confirms that the exponent $n^{1+\frac{1}{r-1}}$ is optimal, completing the asymptotic characterization of the single-error correction threshold for Analog ECCs.

For a graph G let F(G) denote the largest order of a regular induced subgraph of G, and let f(n) = min{F(G) : |V(G)| = n}. A problem of Erdos, Fajtlowicz and Staton asks whether f(n)/log n -> infinity. Dyson and McKay have recently proved f(n) <= (sqrt(2e)+o(1)) sqrt(n) via a type-based random model (arXiv:2604.08215). This paper concerns the opposite direction within the type-based family. We conjecture that every model of the family satisfies F(G) >= (sqrt(2e)-o(1)) sqrt(n) asymptotically almost surely, so that sqrt(2e) is the optimal constant obtainable from the family, and we prove the corresponding statement at exponent level – F(G) >= n^{1/2-eps} – conditionally on two explicitly stated hypotheses: a local limit lower bound for inhomogeneous degree sequences, and a correlation estimate at sublinear overlaps. The complementary overlap range, including full overlap, requires no correlation hypothesis. We further record the exact-curvature first-moment computation that independently identifies sqrt(2e), including a uniform trace bound on its determinant correction, and certified exact computations at orders up to 48 consistent with the predicted crossover F ~ min(n^{2/3}, sqrt(n/L)). This version substantially revises v1; see the note in Section 1.

We prove that the notion of amenability of a (module over) an associative algebra does not depend on the ground field, answering a question proposed by Cornulier in a MathOverflow discussion. A significant part of the argument is based on ideas of ChatGPT 5.6 Sol.

The Zarankiewicz number Z(m,n,s,t) is the maximum number of edges in a bipartite graph with parts of orders m and n containing no copy of Ks,t. We give one combined, certificate-based computer-assisted proof for two finite slices and a corrected neighboring frontier: Z(12,n,3,3) = 6n (18 <= n <= 22), Z(13,22,3,3) = 137, Z(13, 18, 3, 3) = 116, Z(14, 18, 3, 3) = 124, Z(15,18,3,3) = 132, Z(14, 17, 3, 3) = 118, Z(15, 17, 3, 3) = 126, 132 <= Z(16,17,3,3) <= 133. The load-bearing new upper bounds are the exact 12 x 18 and 13 x 18 certificate packages. Their orbit certificates exclude every hypothetical matrix at the next edge count. Deletion lemmas and explicit witnesses close four neighboring cells, while the 16 x 17 entry is deliberately reported as an interval because only its 132-edge lower witness and the published 133 upper bound are certified here. Separately, the 13 x 22 proof excludes 138 ones by reducing to 83 degree profiles, rationally separating 77 of them, and eliminating the remaining six by marked-row congruences, leave enumeration, modular Gram tests, and exact Farkas certificates. All accepted claims are replayed by standard-library Python and exact integer/rational arithmetic; floating-point optimization is used only to discover certificates.

We prove that every countable inner amenable group has vanishing $\ell^2$-Betti numbers in all degrees. This was previously open in every degree $n\geq2$.

Let $f$ be a holomorphic cusp form for $SL(2,\mathbb{Z})$. In this paper, we prove for $T^{\frac{1}{3}} \ll M \ll T^{\frac{1}{2}}$, the following short interval second moment \begin{equation*} \int_T^{T+M}\left|L\left(\frac12+it,f\right) \right|^2 dt \ll_{f,\epsilon} T^\epsilon \left( M + \frac{T}{M}+\sqrt{TM}\right). \end{equation*} The proof uses the trivial delta method along with conductor lowering technique.

In this paper we prove Hölder inequalities for the Besov-Morrey spaces $ \mathcal{N}^{s}_{u,p,q}(\mathbb{R}^{d}) $ and the Triebel-Lizorkin-Morrey spaces $ \mathcal{E}^{s}_{u,p,q}(\mathbb{R}^{d}) $. We observe that these Morrey smoothness spaces are pointwise multiplier spaces if certain conditions concerning the parameters also including $ s > d \max ( 0, \frac{1}{p} - 1) $ are fulfilled. In this context we significantly generalize the Hölder inequalities for the original Besov and Triebel-Lizorkin spaces obtained by Sickel and Triebel in 1995. For the proofs we use paramultiplication and some Franke-Jawerth embeddings obtained by Haroske and Skrzypczak.

Let $G$ be a finite group acting on a Grothendieck category $\mathcal{A}$ with enough projectives, such that $|G|$ is invertible in $\mathcal{A}$. We prove a general lifting theorem for abelian model structures from $\mathcal{A}$ to its equivariant category $\mathcal{A}^G$, and establish a triangle equivalence up to retracts between the corresponding homotopy categories. We also construct a commutative square whose horizontal functors are triangle equivalences and whose vertical comparison functors are triangle equivalences up to retracts. This square relates derived functors on the lifted equivariant model categories to the equivariantizations of the derived functors on the original homotopy categories. In the module category setting, we illustrate the above results using the PGF Hovey triples, and apply them to homotopy squares induced by a Frobenius bimodule and by a stable equivalence of adjoint type.

We present a rigorous error analysis for a semidiscrete immersed CutFEM applied to fluid–structure interaction (FSI) problems involving an incompressible viscous fluid and an elastic solid. The method employs a Lagrange multiplier to enforce the fluid-solid coupling conditions while fully accounting for the effect of the structural deformation on the fluid domain. For sufficiently regular FSI solutions and finite elements of degree $k\geq3$, we prove convergence of order $k$ for the fluid velocity and solid displacement and convergence of order $k-1$ for the fluid pressure in the energy norms associated with the FSI problem. To the best of our knowledge, this is the first rigorous convergence analysis of an unfitted finite element method for a fully coupled FSI system with a deforming interface.

We present a neural network-based method for numerical analytic continuation of solutions of nonlinear partial differential equations (PDEs) and detection of their complex singularities. The proposed framework employs "unsafe" Padé activation units (PAUs) as activation functions, together with a novel backpropagation-free method for computing the weights and biases of the hidden layers. The training procedure is designed specifically for learning meromorphic functions with pole-type singularities. Unlike existing methods with fixed hidden-layer parameters, the proposed approach treats the weights and biases as time-dependent functions, allowing the network to adapt to the evolution of the complex singularities. Using this method, we can efficiently locate the complex singularities of the extended solution, track their trajectories over time, and, based on this dynamics, infer the formation of characteristic phenomena in the solutions of nonlinear PDEs. To demonstrate the performance of our method, we analyze three well-known cases: (1) the nonlinear heat equation, which exhibits finite-time blow up; (2) the nonlinear Burgers equation, which develops a shock; and (3) the nonlinear Schrödinger equation, in which rogue waves form.

There are several methods for searching for graphs with prescribed properties, such as SAT solvers and specialized generators. These methods return the result as raw data: an adjacency matrix or a string encoding. The raw data certifies that the graph exists, but it does not reveal any structural properties of the graph. We ask whether one can automatically discover a short algebraic description if only this raw data is provided. We look for a description such as a Cayley graph $\mathrm{Cay}(\Gamma, S)$ or a lexicographic product $C_5[K_3]$. We address this question with a neurosymbolic approach. We propose an agent that runs on a general-purpose large language model with no fine-tuning or per-target training. The model interleaves reasoning with calls to the computer algebra system SageMath: it analyzes the target graph, proposes and tests candidate constructions, and revises them until the output matches the target. The agent communicates with SageMath through a Model Context Protocol (MCP) server, which we release as a general-purpose bridge. Whether a construction matches the target is checked by a single exact isomorphism test, and therefore rests on the symbolic side and not on the model. We test the approach on a benchmark of 100 highly symmetric graphs, namely two-orbit graphs on up to 25 vertices; the benchmark was fixed in advance. Our agent could find verified algebraic constructions for all of them, without falling back to raw encodings. A strong template-enumeration baseline reaches only about $20\%$, and a catalog lookup could not identify any of these graphs. However, construction quality declines when symmetry is removed. As a concrete application, we identify the smallest known counterexample to the Bernhart-Kainen dispersability conjecture, a $16$-vertex graph that enumeration found as raw data. For this graph, our agent found an explicit algebraic construction.

This paper presents a family of high-resolution weighted essentially non-oscillatory compact least-squares schemes with implicit time integration for the compressible Navier-Stokes equations on curvilinear grids. Compared with the original compact least-squares schemes, the proposed method introduces two main improvements. First, instead of enforcing the accuracy constraints over the entire computational domain, including discontinuous regions, it constructs the reconstruction matrix only along smooth reconstruction lines using an accuracy-preserving weighting strategy. This treatment effectively suppresses the persistent high-frequency oscillations observed in the original compact least-squares schemes and yields sharper profiles near discontinuities. Second, the method simplifies the shock-capturing procedure and improves efficiency by using a single set of polynomials, whereas the original compact least-squares schemes require both unlimited and limited polynomials. Combined with spectral optimization, the proposed method exhibits more favorable spectral properties than conventional weighted essentially non-oscillatory schemes. The smoothness indicators and penalty matrices are constructed through an efficient iterative procedure with modest additional cost. Numerical results for inviscid and viscous one-, two-, and three-dimensional flows demonstrate that the proposed method provides robust shock-capturing capability while maintaining high resolution in smooth regions and across contact discontinuities.

Wireless visual uplinks increasingly carry structured representations for edge inference, making packet reliability part of task-aware link adaptation. In open-vocabulary scene-graph transmission, an indexed triplet is usable only if both its triplet packet and the vocabulary packets defining any newly introduced tokens are recovered. This prerequisite coupling makes the marginal value of packet reliability depend on neighboring packet reliabilities. We formulate a dependency-aware semantic distortion and jointly optimize finite choices of modulation and coding scheme (MCS), transmit power, and Chase-combining hybrid automatic repeat request (HARQ) depth under expected delay and energy constraints. The distortion is multi-affine in packet failure probabilities and cannot, in general, be reduced to static separable unequal error protection (UEP) weights when prerequisites are active. This structure yields a state-dependent reliability coefficient and explicit switching thresholds among wireless actions. A Lagrangian block method performs exact per-packet finite-action updates for fixed multipliers. On reduced instances, it matches exhaustive optimization in 28 of 30 cases, with a worst gap of 1.095%. On GQA traces using a table-driven block error rate (BLER) abstraction, it reduces mean semantic distortion by 58.71% and grounded-query failure by 57.03% relative to dependency-agnostic HARQ under the same budgets.

We develop a perturbative framework for time-dependent point interactions in quantum systems with purely discrete unperturbed spectrum. In two and three dimensions the point interaction is described through the renormalized resolvent obtained from heat-kernel regularization, while in one dimension the diagonal Green function is finite and no renormalization is required. Time dependence is introduced either through the interaction parameter or through the motion of the support point. In both cases we expand the non-autonomous Hamiltonian in the spectral basis of the corresponding static point-interaction problem and derive the first- and second-order pole shifts, projection corrections, and transition amplitudes. We also describe how finite degeneracies are treated by separating the coupled point-interaction direction from the unaffected orthogonal subspace in each degenerate eigenspace. The method is illustrated by explicit examples: point interactions with time-dependent coupling, harmonic oscillators perturbed by moving point interactions, and a particle on a sphere with a moving interaction center. We also discuss the one-dimensional harmonic oscillator with a moving delta potential, showing that in the non-renormalized case the present formulation reduces to the standard time-dependent perturbation theory. Finally, an exactly solvable delta center moving with constant velocity on a circle is used as a discrete-spectrum example for the local moving-center expansion and its velocity-dependent dynamical corrections.

It is a well known result by Singmaster that any integer $d$ divides almost all binomial coefficients. The study of the structure and the size of the exceptional set has been of interest for many decades. In the case where $d$ is a prime power, these sets are well understood and their size is known. However, if $d$ has at least two distinct prime factors, no non-trivial bounds are known. In this paper, we will provide the first non trivial bound on the number of binomial coefficients coprime to 6.