Papers

As a continuation of a previous study of isolation numbers in Cartesian and lexicographic products, we investigate isolation numbers and, more generally, $\cal F$-isolation numbers in direct, strong, lexicographic, and Cartesian products of graphs. For direct products, we derive upper bounds for the $\{K_{n_1,\ldots,n_d}\}$-isolation number in terms of isolation and total domination parameters of the factors, and establish lower bounds based on open packings. We also determine exact values for several infinite families of direct products, including $\iota(P_{4\ell}\times C_{2k+1})=\ell(k+1)$. For strong products, we prove a general lower bound on $\iota(G\,\boxtimes\, H,{\cal F})$ involving the $2$-packing number and provide an upper bound on $\iota(G\,\boxtimes\, H)$. For lexicographic products, we determine the $\cal F$-isolation number in several general settings, obtaining exact formulas in terms of domination and total domination numbers of the first factor. Finally, for Cartesian products, we extend results from our previous work to arbitrary graph families $\cal F$. We introduce $\cal F$-isolation graphs and use $\cal F$-transversals to derive general upper bounds, together with corresponding lower bounds.

In this work, we investigate the inverse problem of recovering the conductivity coefficient in an elliptic equation from noisy measurements collected at finitely many deterministic scattered points in the domain $\Omega$, and corrupted by random noise. Inspired by the regularity analysis, we propose a numerical scheme based on the regularized least-squares formulation with a $W^{1,4}(\Omega)$ penalty, and discretize the regularized problem using the Galerkin finite element method with continuous piecewise linear elements. Under suitable assumptions on the problem data, we provide an error analysis of the regularized solution and its Galerkin approximation. We establish $L^2(\Omega)$ error bounds in a high-probability sense, which depend explicitly on the regularization parameter $\gamma$, the number $n$ of data points and the mesh size $h$. We also present numerical experiments to illustrate the theoretical findings.

For a fixed tree $T$ on $t$ vertices, let $\ex(n,K_r,T)$ denote the maximum number of $r$-cliques in an $n$-vertex $T$-free graph. Write $n=\alpha(t-1)+\beta$ with $0\le\beta<t-1$. A conjecture proposed in extremal-graph form by Alon and Shikhelman~\cite{alon2016}, later stated explicitly as an exact clique-counting formula by Gerbner and Palmer~\cite{GP2026}, asserts that \[ \ex(n,K_r,T)=\alpha\binom{t-1}{r}+\binom\beta r, \] with the extremal construction given by the vertex disjoint union of $\alpha$ copies of $K_{t-1}$ and a graph with $\beta$ vertices. This paper proves this conjecture for every $n$ and for every tree $T$ whenever $C\log t\le r\le t-1$ and $t$ is sufficiently large, where $C$ is an absolute constant. This lowers the previously known general threshold $r\ge\lceil(2t-1)/3\rceil$, which is linear in $t$ to logarithmic order. The paper also proves the conjecture for every $3\le r\le t-1$ when at least $\lceil(t-3)/2\rceil$ leaves of $T$ have a common neighbor, and this extends the known results for stars and double stars.

A key result in statistics is the data processing inequality, originally proved by Blackwell (1951) and later refined by DeGroot (1962) in terms of statistical uncertainty. The latter statement claims that the Bayes risk achieved on raw data always undercuts the Bayes risk on a processed version of the same data. This seemingly contradicts empirical findings in machine learning: pre-training, representation learning, feature learning, data augmentation are techniques used to improve performance of machine learning models. We reconcile both worlds by simply accepting that machine learning problems are constrained learning problems: the model class used does not include all measurable functions. We present counterexamples showing that the classical data processing inequality fails to hold in such a setting. Hence, we formulate a generalized data processing inequality, requiring the constrained Bayes risk of a joint distribution (with respect to a loss function and a constrained model class) to lower bound the constrained Bayes risk on the stochastically modified data distribution, regardless of the choice of distribution. We show this inequality to be equivalent to a set containment condition on a specific function set induced by the loss and model class, called the superprediction set. Finally, we exploit our characterization, derive sufficient conditions for this containment and quantify the inequality-gap.

Estimation of a regression function from independent and identically distributed data is considered. The $L_2$ error with integration with respect to the design variable is used as the error criterion. An initially randomly pruned fully connected deep neural network with logistic squasher as activation function is fitted to the data via gradient descent, using a data-dependent choice of non-constant stepsizes during gradient descent. It is shown that this network achieves (up to a logarithmic factor) the optimal minimax rate of convergence in case that the regression function is $(p,C)$–smooth. Here the estimate is able to circumvent the curse of dimensionality provided the predictors are concentrated in the neighborhood of a low dimensional manifold. The finite sample size performance of the estimate is illustrated by applying it to the simulated data.

Barría and Halmos [Trans. Amer. Math. Soc. (1982), (10) Example] asked whether the Cesàro operator belongs to the Toeplitz algebra. We answer this question affirmatively and prove that the Cesàro operator is the product of a Toeplitz operator and a continuous function of a self-adjoint Toeplitz operator. Rhaly operators are a natural generalization of the Cesàro operator. We give necessary and sufficient conditions for a Rhaly operator to belong to the Toeplitz algebra or the Hankel algebra. The main approach is to use the Fejér kernels to get localized estimates for operators. This yields a necessary condition for an operator to belong to the Hankel algebra. As an application, we find a class of Rhaly operators that are asymptotic Toeplitz operators and essentially commute with the unilateral shift, but are not in the Hankel algebra. This answers (17) Question in the same paper.

Multivariable Alexander theory produces Fitting ideals without preferred generators. After fixing a coefficient field and a monomial order, we represent these ideals by reduced Gröbner bases of their polynomial contractions, obtaining Alexander–Gröbner invariants of links. Over \(\mathbb Q\), for the maximal free-abelian coefficient system of a connected compact oriented \(3\)-manifold whose nonempty boundary is a disjoint union of tori, we deduce determinant-divisor reciprocity from Blanchfield duality. We compute these invariants for torus links and low-crossing links, and we determine \(\AG_2\) for a two-component pretzel family.

We establish an asymptotic formula for the mixed second moment involving the second derivative of the completed $L$-function, when averaged over monic, irreducible polynomials in the function field setting.

Lower bounds on the pointwise spatial decay of the ground state of the Pauli-Fierz model in non-relativistic quantum electrodynamics are studied. Two models are considered: the full Pauli-Fierz model and the Pauli-Fierz model under the dipole approximation. For the full model, a lower bound is derived by means of a probabilistic approach based on the Feynman-Kac formula. The application of the Agmon distance method to the full model encounters a serious difficulty arising from the path dependence of a double stochastic integral. In contrast, under the dipole approximation, the corresponding integrand becomes deterministic, which allows a lower bound to be obtained in terms of the Agmon distance.

Estimation of a regression function from exponentially $\beta$-mixing data is considered. The $L_2$ error with integration with respect to the design is used as the error criterion. Deep neural network estimates with logistic activation function are defined, where all parameters are learned by gradient descent. The rate of convergence of the expected $L_2$ error is analyzed for $(p,C)$-smooth regression functions. In the special case that the design is concentrated on a $d^*$-dimensional manifold, it is shown that the expected $L_2$ error of the estimate achieves a rate of convergence which depends on $d^*$ and not on the dimension $d$ of the design.

Quadratic ball discrepancy measures how uniformly a code meets Hamming balls over all centers and radii. Perfect codes minimize this quantity among codes of the same cardinality. We prove a quantitative refinement: at the parameters of every perfect code other than a one-word code or the full space, the discrepancy excess over the minimum is at least an explicit positive multiple of the discrepancy at the packing radius. The latter equals the tiling defect divided by the square of the code cardinality. This defect is the total squared deviation of the covering multiplicity from one. The comparison applies to all codes of that cardinality. For one-error perfect codes, the coefficient is at least one, and no larger constant holds for all such codes; explicit coefficients depending on the parameters can be much larger. The tiling defect also measures the failure of uniform ball noise to smooth a code distribution. Consequently, the same discrepancy excess gives explicit bounds on holes, overlaps, and divergence from the uniform distribution.

In the current work, we study univariate polynomial weighted histopolation on $[-1,1]$, where the data are weighted integrals over a family of intervals. After choosing a polynomial basis, the weighted moment conditions lead to a histopolation matrix whose structure depends on the weight and on the geometry of the cells. We investigate its nonsingularity, which guarantees unisolvence, together with exact diagonality of its Gram matrix, which allows its singular values and spectral condition number to be determined explicitly. For families of intervals whose endpoints belong to a fixed grid, we characterize unisolvence in terms of the connectedness of the associated endpoint graph. In the unisolvent case, this graph is a tree, and the unique paths joining consecutive grid points provide an explicit expression for the inverse matrix. This identity gives explicit formulas for the singular values of both matrices, and shows that their condition numbers in the two-norm coincide and grow linearly with the matrix size. Moreover, it yields the limiting singular value distributions of the two matrix sequences. We also establish a general diagonalization criterion based on discrete weighted orthogonality. The criterion recovers the first kind Chebyshev construction and leads to a diagonal configuration for the constant weight based on discrete sine orthogonality. For interval families with a connected endpoint graph, the corresponding moment vectors define an inner product on the polynomial space and lead to a monic basis with a diagonal weighted Gram matrix. Finally, we derive reduction formulas for cell moments associated with generalized Jacobi weights and introduce an alternative basis for shifted Jacobi weights. Applied to the Chebyshev weight of the fourth kind, this basis, together with a correction of one nonconstant element, yields an exactly diagonal Gram matrix.

We prove that every compact four-dimensional weakly Einstein Ricci soliton is Einstein. The nontrivial compact case reduces to the gradient shrinking setting, where a differential identity for weakly Einstein four-manifolds, together with the curvature identity for gradient Ricci solitons, yields the pointwise relation $|R|^2\nabla f=0$ for the soliton potential $f$. Consequently, no compact proper weakly Einstein four-manifold admits a Ricci soliton structure. A noncompact homogeneous example shows that the compactness assumption is essential.

We study the quasilinear Schrödinger equation \begin{align*} -\Delta_{\mathbb{H}} u +V(\xi)u-\Delta_{\mathbb{H}} (\left|u\right|^{2\alpha})\left|u\right|^{2\alpha-2} u= \lambda \left|u\right|^{q-2}u + \left|u\right|^{p-2}u \quad \text{ in } \mathbb{H}^N, \end{align*} where $\Delta_{\mathbb{H}}$ is the Kohn Laplacian on the Heisenberg group $\mathbb{H}^N$, $4\alpha<q<p \leq 2\alpha Q^*$, $\alpha>\frac12$, and $2\alpha Q^{*}$ is the critical exponent, $Q=2N+2$ being the homogeneous dimension and $Q^{*}=\frac{2Q}{Q-2}$. For $p=2\alpha Q^{*}$ and $\lambda>0$, we obtain a nontrivial solution, assuming that the potential is bounded below by a positive constant and is either asymptotically constant from above or invariant under a discrete subgroup of $\mathbb{H}^N$. In the opposite direction, we prove a Pohožaev identity for $\Delta_{\mathbb{H}}$ and combine it with the Nehari identity, obtaining a family of identities from which the quasilinear energy disappears exactly at the exponent $2\alpha Q^{*}$. This yields a nonexistence theorem under a monotonicity condition on the potential with respect to anisotropic dilations and shows that no nontrivial solution exists for $\lambda \leq 0$; the sign of the subcritical perturbation determines solvability. Along the way, we show that every weak solution is bounded and decays exponentially in the Korányi gauge.

We present a framework for studying the moduli space of linear determinantal representations for flat families of complex projective cubic surfaces across singularity boundaries. First, we deal with the classical deformation theory where a surface S_0 possesses a rational double point (RDP) of type E_6. By establishing a global simultaneous resolution over a finite ramified Galois covering of the global parameter slice, we realize the relative moduli space H_N as a diagonal Weyl quotient (h x R)/W(E_6). Second, we extend this classification to the critical boundary configuration where S0 contains a unique isolated simple elliptic singularity of type E~_6. Because the local monodromy group becomes infinite, the classical simultaneous resolution framework completely breaks down. We bypass this obstruction by constructing a geometric substitute Z derived from the filtration of Looijenga's invariant algebra of affine Weyl groups. We also show that the vector bundle Z decomposes into a direct sum of line bundles over the structural elliptic curve E, with degrees explicitly matching the negative Coxeter marks of the highest root of E_6. Utilizing Riemann's extension theorem and vector bundle rigidity over elliptic curves, we settle our isomorphism result, proving that the moduli space H-bar representing the semiuniversal family is globally isomorphic to the total space of Z.

We propose and analyze a first-order numerical scheme for a class of scalar Itô stochastic differential equations with almost surely positive solutions. We construct a new explicit numerical scheme, such that for any choice of the time-step size, the numerical solution is guaranteed to remain almost surely positive. The main result of this article is the first-order strong convergence of the proposed positivity preserving scheme, which is illustrated with numerical experiments and proved rigorously. It is also shown how to adapt the scheme and the results to Stratonovich stochastic differential equations with positive solutions.

We give an example of a coherent theory, a topological space, and a continuous assignment of the (model-theoretic) types of the theory to the points of the space, such that there is no sheaf model of the theory over the space with the property that the fibre over a point, as a set-based model, realises the corresponding type; thus, we answer in the negative a problem posed by Kristóf Kanalas.

We construct and analyze numerical schemes for systems of stochastic differential equations, which preserve almost surely a given hypercube of arbitrary dimension. We propose a new general class of explicit schemes, such that for any choice of the time-step size the numerical solution takes values in the hypercube. We prove strong and weak convergence results for this general class of domain preserving numerical schemes, with strong order $1/2$ and weak order $1$ in general. We also construct a variant of the scheme which achieves strong order $1$ when the stochastic differential equation is driven by a one-dimensional Brownian motion. The convergence results are illustrated with numerical experiments.

In this paper, we prove a product rule for Goodwillie derivatives: given a differentiable $\infty$-category $\mathcal{C}$ whose stabilization is equivalent to the $\infty$-category $\mathrm{Sp}$ of spectra, we show that the derivatives functor $\partial_* \colon \mathrm{Fun}^\omega(\mathcal{C}, \mathrm{Sp}) \to \mathrm{RMod}_{\partial_*\mathrm{id}_\mathcal{C}}(\mathrm{SSeq}(\mathrm{Sp}))$ is strong symmetric monoidal, where the source is equipped with the pointwise tensor product and the target with Day convolution. Since the Koszul dual of $\partial_*\mathrm{id}_\mathcal{C}$ can be recovered as a coendomorphism operad from Day convolution, this product rule is useful for calculating the operad $\partial_*\mathrm{id}_\mathcal{C}$ in examples. We derive the product rule as a consequence of the more general statement that taking derivatives preserves cartesian products on the $(\infty, 2)$-categorical level. In fact, the main theme of this paper is that the extraction of Goodwillie derivatives preserves a lot of structure when regarded as a functor of $(\infty, 2)$-categories: apart from products, it also preserves cotensors and certain pullbacks. We illustrate how our results can be used to calculate Goodwillie derivatives by determining the operad structure on the derivatives of the identity functor in pointed spaces, algebras over an operad and sheaves on a site.

We construct the half-signature bordism bicategory $\mathbf{Bord}_{1,2,3}^\mathrm{sig/2}$ as a central extension of the 3-dimensional oriented bordism bicategory by $\mathbb{Z}$. This is an index $2$ subextension of the signature bordism bicategory. Following methods of Bartlett–Douglas–Schommer-Pries–Vicary, we construct a presentation of this bicategory and hence show that its linear representations are classified exactly by modular tensor categories.

We introduce a computational method that extends the range of current global ocean climate models to non-hydrostatic scales, and establishes a convection resolving ocean model. The main obstruction towards this goal at global scale -the calculation of the non-hydrostatic pressure- is resolved by computing the pressure locally in a manner that capitalizes on particular conditions of ocean dynamics, and the structure of ocean models. Our claims are substantiated 1) by a theoretical analysis, which shows that the proposed method is compatible with ocean physics, 2) by numerical experiments, which evidence by comparison that it represents faithfully and to high accuracy non-hydrostatic dynamics, and 3) by a performance analysis that establishes that our non-hydrostatic method exceeds the cost of a hydrostatic model by a fixed factor of about $1.2$ in operation count and $1.3$ in runtime at any resolution. We show how to use the additional information contained in non-hydrostatic scales can be used to calculate the diffusivity free from numerical contributions and the mixing efficiency, transforming both from prescribed values by the modeller to calculated values from the dynamics.

We establish a grading bound for colored $\mathfrak{gl}_N$ link homology that can be read off from any link diagram. In the context of contact geometry, this gives a bound on the Thurston–Bennequin number of a Legendrian link from $\mathfrak{gl}_N$ link homology, in the spirit of Ng.

We consider the radially symmetric version of the reaction-diffusion equation $u_t-d\Delta u=f(u)$ with a monostable nonlinearity $f$, viewed as a model for the spreading of a species with population range $r<h(t)$ and density $u(t,r)$ ($r=|x|$), where the free boundary $r=h(t)$ is governed by $u(t,h(t))=\delta>0$ and $h'(t)=-d u_r(t,h(t))/\delta$. For the one-dimensional case ($N=1$), Du \cite{DN} proved that when $\delta\in(0,1)$, spreading occurs: $u\to1$ locally uniformly in $\mathbb{R}$, $h(t)\to\infty$, and $\lim_{t\to\infty}[h(t)-c_*t]=\tilde{h}\in\mathbb{R}$ with no logarithmic shift. In the present paper we consider $N\ge2$ and establish a complete trichotomy: spreading for $\delta\in(0,1)$; transition for $\delta=1$, where $u\to1$ uniformly on $[0,h(t)]$ and $h(t)\to h_\infty\in(0,\infty)$; and vanishing for $\delta>1$, where $h(t)\to0$ and $u\to\delta$ uniformly on $[0,h(t)]$. For the spreading regime, by constructing sharp upper and lower solutions, we prove that the solution converges globally to the semi-wave profile and reveal a logarithmic shift of the form $ \lim_{t\to\infty}\big[h(t)-c_*t+c_N(\delta)\log t\big]=\hat{h}\in\mathbb{R}$, with the coefficient $c_N(\delta)>0$ satisfying $ \lim_{\delta\to0}c_N(\delta)=d(N-1)/c_0$, where $d(N-1)/c_0$ is the shift coefficient for the high-dimensional radial pushed-case Cauchy problem. These results reveal the connection to the spreading behavior modeled by the corresponding Cauchy problem.

We establish a new concavity inequality for the elementary symmetric function \(\sigma_k\), which controls the spectral quadratic form arising from the second variation of \(\log\lambda_{\max}\). The proof is based on an easy–hard decomposition of the spectral variables. The easy region is treated using an optimal constrained concavity estimate for \(\sigma_k\), whereas the hard region is analyzed through Gårding-root coordinates and the concavity of the ordered partial sums of the inverse roots. As an application, for \(n/2\leq k<n\), we obtain global curvature estimates for closed star-shaped \(k\)-convex hypersurfaces satisfying \(\sigma_k(\kappa)=f(X,\nu)\) with a general positive right-hand side, together with the corresponding global-to-boundary estimates for Euclidean Hessian equations.

We introduce stable symmetric series which encode normalized conjugacy classes and their multiplication operators simultaneously for all symmetric groups. This gives a direct route from the Ivanov–Kerov algebra to shifted symmetric functions and to differential operators in $U(\mathcal W_{1+\infty})$. Using the Goulden–Jackson product, we extend the construction to Jack polynomials, recover shifted Jack eigenvalues and Pieri-type relations, and obtain explicit candidate operators in degrees three and four. Their real and quaternionic specializations to zonal polynomials are verified by Gaussian matrix integrals and exhaustive Wick enumeration.

We construct certain special rational elliptic surfaces with four reduced singular fibres that are naturally attached to the platonic solids and belong to a group of surfaces complementing the well-known semi-stable Beauville surfaces. We describe the basic algebraic geometric properties of these surfaces, their associated Picard-Fuchs operators, integer sequences, Laurent polynomial representations and Apéry constants. On the arithmetic side, we discover the need for a refinement of the Dwork-expansion used to find the Euler factors of the fibres of these fibrations. This is explained by comparing the $q$-coordinates coming from the elliptic curve and the $q$-coordinate of the Picard-Fuchs equation and is also directly reflected in the shape of the monodromy matrices in the Frobenius basis.

For every odd integer $d\geq3$, a continuous function $\varphi\colon[0,\infty)\to\mathbb R$ supported in $[0,\pi]$ and isotropic positive definite on $\mathbb R^d$ remains so on $\mathbb S^d$. In even dimensions, recent work shows that this transfer fails under every prescribed positive upper bound on the support. We prove an exact-support refinement with a construction uniform in the prescribed radius. More precisely, for each $d=2m\geq2$ and $R\in(0,\pi]$, we construct a function $\varphi$ whose radial extension belongs to $C_c^\infty(\mathbb R^d)$ and has support radius exactly $R$, such that $\varphi(\|\mathbf{x}-\mathbf{y}\|_2)$ is strictly positive definite on $\mathbb R^d$, whereas $\varphi(\rho(\mathbf{x},\mathbf{y}))$ is not positive definite on $\mathbb S^d$. Thus every admissible support radius is attained by a smooth, strictly Euclidean positive-definite counterexample.

Let $\mathbb{Z}[1/n] := \{a/n^r: a \in \mathbb{Z},r \in \mathbb{Z} \geq 0\}$ be the euclidean domain obtained from the ring of integers $\mathbb{Z}$ by localizing at $n$. Moreover, let $\text{SL}_2(\mathbb{Z}[1/n])$ be the group of all invertible 2-by-2 matrices of determinant one with entries in $\mathbb{Z}[1/n]$. The homology groups $H_k(\text{SL}_2(\mathbb{Z}[1/n]),\mathbb{Z})$ are of interest in Geometric Group Theory, Algebraic Number Theory and Algebraic K-theory. In this dissertation, we study these groups via a spectral sequence derived by the action of $\text{SL}_2(\mathbb{Z}[1/n])$ on the product of the trees $B_p$ associated with the $p$-adic valuation on $\mathbb{Q}$ where $p$ is a prime factor of $n$.

Graph burning is a round-based process which can be viewed as a discrete one-player game that models the spread of influence throughout a network. Extending this process to hypergraphs can be done in numerous ways; two such processes that have been studied include the burning process and the lazy burning process for hypergraphs. Combinatorial designs can be thought of as hypergraphs with useful and interesting characteristics. In this paper we explore the burning and lazy burning processes in Steiner triple systems (STSs). We obtain logarithmic bounds on the burning number of an arbitrary STS and prove the existence of an STS with burning number $\rho$ for any integer $\rho \geq 3$. We observe that the concept of lazy burning is equivalent to the notion of the dimension of an STS, and we consider ramifications of this equivalence. We also show that the difference between the burning and lazy burning number of a Steiner triple system can be arbitrarily large. Finally, we consider burning and lazy burning numbers of affine and projective triple systems in detail.

We show that a contravariantly finite resolving subcategory over a henselian local ring must be the subcategory of free modules or the whole module category or the subcategory of maximal Cohen-Macaulay modules. This removes the Gorenstein assumption from a theorem of Takahashi. We also show that a resolving subcategory of finite type other than the subcategory of free modules must be the subcategory of maximal Cohen-Macaulay modules. As a consequence, every Cohen-Macaulay local ring of finite CM type is uniformly dominant, thereby establishing a stronger form of a conjecture of Takahashi.