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arXiv:1709.08464v3 Announce Type: replace Abstract: We study the controllability of linear control systems on connected solvable Lie groups with hyperbolic drift. Under a suitable splitting assumption on the control directions, we show that the controllability problem can be reduced to the controllability of induced systems on the positive and negative hyperbolic components of the group. We then establish sufficient conditions ensuring con..
arXiv:1902.07030v2 Announce Type: replace Abstract: Global sensitivity analysis (GSA) of numerical simulators aims at studying the global impact of the input uncertainties on the output. To perform the GSA, statistical tools based on inputs/output dependence measures are commonly used. We focus here on dependence measures based on reproducing kernel Hilbert spaces: the Hilbert-Schmidt Independence Criterion denoted HSIC. Sometimes, the pro....
arXiv:1904.09888v3 Announce Type: replace Abstract: Penney's game is a two-player zero-sum game in which each player chooses a three-flip pattern of heads and tails, and the winner is the player whose pattern occurs first in repeated tosses of a fair coin. Because the players choose sequentially, the second mover has the advantage. In fact, for any three-flip pattern, there is another three-flip pattern that is strictly more likely to occu..
arXiv:2003.11419v2 Announce Type: replace Abstract: In this article, we introduce incomplete Lauricella matrix functions (ILMFs) of $n$ variables through application of the incomplete Pochhammer matrix symbols. Furthermore there is derivation of certain properties; matrix differential equation, integral formula, recursion formula and differentiation formula of the ILMFs. We also establish the connection between these matrix functions and o..
arXiv:2005.06813v3 Announce Type: replace Abstract: We provide a proof that the classes of finitely generated Kleinian groups and of three-manifold groups are quasi-isometrically rigid.
arXiv:2006.13218v3 Announce Type: replace Abstract: In 2011 Musiker, Schiffler and Williams obtained expansion formulae for cluster algebras from orientable surfaces. For singly and doubly notched arcs these formulae required the notion of $\gamma$-symmetric perfect matchings and $\gamma$-compatible pairs of $\gamma$-symmetric perfect matchings, respectively. We simplify and unify these approaches by considering good matchings of loop grap..
arXiv:2010.06759v4 Announce Type: replace Abstract: This paper presents an algorithmic study and complexity analysis for solving distributionally robust multistage convex optimization (DR-MCO) problems. Our main contribution is a novel nonconsecutive dual dynamic programming (NDDP) algorithm which explores different stages in an adaptive fashion. In contrast with the usual consecutive dual dynamic programming (CDDP) algorithm, we show that..
arXiv:2102.03384v5 Announce Type: replace Abstract: In this paper, we prove several Ax-Schanuel type results for uniformizers of geometric structures; our general results describe the differential algebraic relations between the solutions of the partial differential equations satisfied by the uniformizers. In particular, we give a proof of the full Ax-Schanuel Theorem with derivatives for uniformizers of simple projective structure on curv..
arXiv:2104.04408v3 Announce Type: replace Abstract: Let $f$ be a Laurent polynomial in $d$ commuting variables with integer coefficients. Associated to $f$ is the principal algebraic $\mathbb{Z}^d$-action $\alpha_f$ on a compact subgroup $X_f$ of $\mathbb{T}^{\mathbb{Z}^d}$ determined by $f$. Let $N\ge1$ and restrict points in $X_f$ to coordinates in $N\mathbb{Z}^d$. The resulting algebraic $N\mathbb{Z}^d$-action is again principal, and is....
arXiv:2107.14176v5 Announce Type: replace Abstract: In this paper we define and compare several new Quillen model structures which present the homotopy theory of algebraic quantum field theories. In this way, we expand foundational work of Benini et al. by providing a richer framework to detect and treat homotopical phenomena in quantum field theory. Our main technical tool is a new extension model structure on operadic algebras which is c..
arXiv:2112.14728v2 Announce Type: replace Abstract: Let $X$ be a smooth projective curve over a complete discretely valued field $K$. Let $L/K$ be the minimal extension such that $X \times_K L$ has a semi-stable model, and write $e(L/K)$ for the ramification index of $L/K$. Let $e(X)$ be the so-called ``stabilization index'' of $X$, defined by Halle and Nicaise as the lcm of the multiplicities of the ``principal'' irreducible components of..
arXiv:2204.11261v3 Announce Type: replace Abstract: We study the Klein-Gordon equation with general interaction terms, which may be linear or nonlinear, and space-time dependent. We initiate the study of such equations with large (non-radial) data. We prove that global solutions are asymptotically given by a free wave and a weakly localized part. The proof is based on constructing in a new way the Free Channel Wave Operator, and further to..
arXiv:2206.15361v3 Announce Type: replace Abstract: The purpose of this article is to give a new construction of the map relating the Borel-Serre and the Baily-Borel compactifications of a Shimura variety (Zucker 1983), and to provide a close analysis of its main properties.
arXiv:2208.00336v2 Announce Type: replace Abstract: We prove irreducible components of moduli spaces of semistable representations of skewed-gentle algebras, and more generally, clannish algebras, are isomorphic to products of projective spaces. This is achieved by showing irreducible components of varieties of representations of clannish algebras can be viewed as irreducible components of skewed-gentle algebras, which we show are always n..
arXiv:2208.09128v5 Announce Type: replace Abstract: We study the totally nonnegative part of the complete flag variety and of its tropicalization. We show that Lusztig's notion of nonnegative complete flag variety coincides with the flags in the complete flag variety which have nonnegative Pl{\"u}cker coordinates. This mirrors the characterization of the totally nonnegative Grassmannian as those points in the Grassmannian whose Pl{\"u}cker..
arXiv:2208.13855v4 Announce Type: replace Abstract: We study rigidity without assuming general position. Given $n$ distinct labelled points and a set $\mathcal{P}\subseteq \binom{[n]}{2}$ of revealed pairs, we ask when the corresponding distances determine the configuration up to isometry. On the line, we prove an extremal result: if $|\mathcal{P}|=\Omega(n^{3/2})$, then there is an induced globally rigid subgraph on $\Omega(|\mathcal{P}|/....
arXiv:2209.07720v4 Announce Type: replace Abstract: In this note, we explore the cohomological property of the codimension of the center of a resolution. In particular, we define a resolution $f:X'\to X$ to be $q$-birational if the center of $f$ satisfies $\mathrm{codim} \,\mathrm{Cent}(f)\ge q+1$, and we prove that $R^if_*\mathcal{O}_X(E)=0$ for every $1\le i\le q-1$ and every $f$-anti-nef effective $f$-exceptional divisor $E$ on $X'$ if ..
arXiv:2303.10146v3 Announce Type: replace Abstract: We introduce a new cohomology theory for stacks called elliptic Hochschild homology, prove some fundamental properties and compute it in some classes of examples. We then introduce its periodic cyclic version and show that, over the complex numbers and for a quotient stack, this recovers Grojnowski's equivariant elliptic cohomology of the analytification.
arXiv:2307.02816v2 Announce Type: replace Abstract: We prove that for every planar graph $X$ of treedepth $h$, there exists a positive integer $c$ such that for every $X$-minor-free graph $G$, there exists a graph $H$ of treewidth at most $f(h)$ such that $G$ is isomorphic to a subgraph of $H\boxtimes K_c$. This is a qualitative strengthening of the Grid-Minor Theorem of Robertson and Seymour (JCTB 1986), and treedepth is the optimal param..
arXiv:2307.08497v3 Announce Type: replace Abstract: It is an easy observation that if a graph~$G$ admits a path-decomposition whose parts have small radius, then $G$ contains no large subdivision of $K_{1,3}$ or $K^3$ as a (quasi-)geodesic subgraph. We show that these are in fact the only obstructions to such path-decompositions of small radial width, and we prove analogous results for decompositions modelled on cycles and subdivided stars..
arXiv:2307.16736v3 Announce Type: replace Abstract: Let $F$ be a totally real number field and $r=[F :\mathbb{Q}].$ Let $A_k(\mathfrak{N},\omega) $ be the space of holomorphic Hilbert cusp forms with respect to $K_1(\mathfrak{N})$, of weight $k=(k_1,\dots,k_r)$ such that $k_j>2$ for all $j$, and with central Hecke character $\omega$. For integral ideals $\mathfrak{N}$ and $\mathfrak{n}$ in $F$ such that $( \mathfrak{n}, \mathfrak{N}) = 1$,....
arXiv:2308.07591v4 Announce Type: replace Abstract: In this paper, for Markov Decision Processes (MDPs) with standard Borel spaces, (i) we first provide a discretization based approximation method for MDPs with continuous spaces under average cost criteria, and provide error bounds for approximations when the dynamics are only weakly continuous (for asymptotic convergence of errors as the grid sizes vanish) or Wasserstein continuous (with ....
arXiv:2308.16614v3 Announce Type: replace Abstract: For each $\mathbf{k}\in\mathbb{C}^4$, let $V_\mathbf{k}$ be the character variety on a four-holed sphere and $\Gamma_\mathbf{k}$ the group generated by the Vieta involution maps. First, under a certain condition, we find a fundamental domain for $\Gamma_\mathbf{k}$-action on $V_\mathbf{k}$, expressed via inequalities. Second, we show that it is decidable whether or not two integral soluti..
arXiv:2309.15384v3 Announce Type: replace Abstract: We give an explicit characterization of the standard monomials for positroid varieties with respect to the Hodge degeneration and give a Gr\"obner basis. Furthermore, we show that promotion and evacuation biject standard monomials of a positroid variety with those of its cyclic shifts and $w_0$-reflection, respectively. The connection to promotion allows us to identify standard monomials ..
arXiv:2310.20517v2 Announce Type: replace Abstract: We study hyperuniformity of self-similar point processes arising from substitution rules in two dimensions. In particular, we derive a sufficient condition for hyperuniformity of these point processes only in terms of the associated substitution matrix. This condition applies to a wide class of examples for which hyperuniformity had not yet been established, including most well-known exam..
arXiv:2311.11190v2 Announce Type: replace Abstract: The set of partial partitions of $\{1,\ldots,n\}$, ordered by containment, forms an abstract simplicial complex $D_n$ whose vertices are the nonempty subsets of $\{1,\ldots,n\}$ and whose simplices are collections of pairwise disjoint subsets. We prove that $D_n$ is vertex-decomposable, give an explicit nonpure shelling, and use it to compute the reduced homology: for $1 \le j \le n$, the..
arXiv:2312.00211v3 Announce Type: replace Abstract: The goal of this paper is twofold; on one hand we wish to present some statements that can be formulated in terms of Interpolation theory which are equivalent to the truth or the falseness of the Riemann Hypothesis, on the other hand we will use a key result of the Jawerth-Milman extrapolation to improve on a well-known criterion by Beurling and Nyman for the Riemann Hypothesis giving sha..
arXiv:2312.07726v3 Announce Type: replace Abstract: We study hitting times in simple random walks on graphs, which measure the time required to reach specific target vertices. Our main result establishes a sharp lower bound for the variance of hitting times. For a simple random walk on a graph with $n$ vertices, we prove that the variance of the hitting time from a vertex $x$ to a vertex $y$, denoted $\tau_y$, is at least of the order $\ma..
arXiv:2402.02247v3 Announce Type: replace Abstract: Numerical approximations of Landau-type operators represent fundamental components of time integration methods for demanding problems such as inhomogeneous Vlasov-Landau-type equations. Substantial computational issues arise from the treatment of the physically most relevant three-dimensional case with Coulomb-type interaction. This work is concerned with the introduction and numerical co....
arXiv:2402.03150v3 Announce Type: replace Abstract: Chv\'{a}tal conjectured that amongst the largest intersecting subfamiles of a finite subset-closed family of sets is a star. Kleitman later strengthened Chv\'{a}tal's conjecture, defining a partial ordering on the vector space freely generated by $2^{[n]}$ and suggesting that the vector of every maximal intersecting subfamily of $2^{[n]}$ is bigger than a convex combination of stars. We r..
arXiv:2402.12471v5 Announce Type: replace Abstract: Cosymplectic and normal almost contact structures are analogues of symplectic and complex structures that can be defined on 3-manifolds. Their existence imposes strong topological constraints. Generalized geometry offers a natural common generalization of these two structures: $B_3$-generalized complex structures. We prove that any closed orientable 3-manifold admits such a structure, whi..
arXiv:2402.14969v2 Announce Type: replace Abstract: In a recent paper, the author and St\"ohr established a bound on the number of iterated Frobenius pullbacks needed to transform a non-smooth purely inseparable point on a regular geometrically integral curve into a rational point. In this paper we improve this result, by establishing a new bound that is sharp in every characteristic $p>0$.
arXiv:2403.20166v2 Announce Type: replace Abstract: The $\varepsilon$-boundary of a set ${A}\subseteq\mathbb{R}^2$ is the set $\{{p}\in\mathbb{R}^2:\rho({p},{A})=\varepsilon\}$, where $\rho$ is the Euclidean distance. We prove that if ${A},{B}\subseteq\mathbb{R}^2$ are nonempty, connected sets, ${A}$ is bounded, and $0<\varepsilon<\rho({A},{B})$, then the $\varepsilon$-boundary of ${A}$ contains a simple closed curve (aka a Jordan curve) t....
arXiv:2404.03950v3 Announce Type: replace Abstract: Given a matching $M$ in the hypercube $Q^n$, the \emph{profile} of $M$ is the vector $\boldsymbol{x}=(x_1,\ldots, x_n) \in \mathbb{N}^n$ such that $M$ contains $x_i$ edges whose endpoints differ in the $i$th coordinate. If $M$ is a perfect matching, then it is clear that $||\boldsymbol{x}||_1 = 2^{n-1}$ and it is easy to show that each $x_i$ must be even. Verifying a special case of a con..
arXiv:2404.10184v2 Announce Type: replace Abstract: We prove an accessibility theorem for finite-index splittings of groups. Given a finitely presented group G there is a number n(G) such that, for every reduced locally finite G-tree T with finitely generated stabilizers, T/G has at most n(G) vertices and edges. We also show that deformation spaces of locally finite trees (with finitely generated stabilizers) are maximal in the partial ord..
arXiv:2406.03860v3 Announce Type: replace Abstract: We introduce a new definition of a model for a formal mathematical system. The definition is based upon the substitution in the formal systems, which allows a purely algebraic approach to model theory. This is very suitable for applications due to a general syntax used in the formal systems. For our models we present a new proof of the downward L\"owenheim-Skolem Theorem for elementary su..
arXiv:2407.06865v4 Announce Type: replace Abstract: We provide a geometric realization of the quasi-split affine $\imath$quantum group of type AIII$_{2n-1}^{(\tau)}$ in terms of equivariant K-groups of non-connected Steinberg varieties of type C. This uses a new Drinfeld type presentation of this affine $\imath$quantum group which admits very nontrivial Serre relations. We then construct \`a la Springer a family of finite-dimensional stand..
arXiv:2407.14048v3 Announce Type: replace Abstract: We introduce a new family of higher-rank graphs, whose construction was inspired by the graphical techniques of Lambek \cite{Lambek} and Johnstone \cite{Johnstone} used for monoid and category emedding results. We show that they are planar $k$-trees for $2 \le k \le 4$. We also show that higher-rank trees differ from $1$-trees by giving examples of higher-rank trees having properties whic..
arXiv:2407.16261v3 Announce Type: replace Abstract: We establish a martingale-type characterisations for the continuum Gaussian free field (GFF) and for fractional Gaussian free fields (FGFs), using their connection to the stochastic heat equation and to fractional stochastic heat equations. The main theorem on the GFF generalizes previous results of similar flavour and the characterisation theorems on the FGFs are new. The proof strategy ..
arXiv:2408.16656v2 Announce Type: replace Abstract: We develop a Sequential Quadratic Optimization (SQP) algorithm for minimizing a stochastic objective function subject to deterministic equality constraints. The method utilizes two different stepsizes, one which exclusively scales the component of the step corrupted by the variance of the stochastic gradient estimates and a second which scales the entire step. We prove that this stepsize ....
arXiv:2409.15532v3 Announce Type: replace Abstract: Stochastic differential equations are ubiquitous modelling tools in physics and the sciences. In most modelling scenarios, random fluctuations driving dynamics or motion have some non-trivial temporal correlation structure, which renders the SDE non-Markovian; a phenomenon commonly known as ``colored'' noise. Thus, an important objective is to develop effective tools for mathematically an....
arXiv:2410.03922v2 Announce Type: replace Abstract: Upon almost-every realisation of the Brownian continuum random tree (CRT), it is possible to define a canonical diffusion process or `Brownian motion'. The main result of this article establishes that the cover time of the Brownian motion on the Brownian CRT (i.e.\ the time taken by the process in question to visit the entire state space) is equal to the infimum over the times at which th....
arXiv:2410.05812v2 Announce Type: replace Abstract: Our objective is to explore random walks on the general linear group, constrained to a specific domain, with a primary focus on establishing the conditioned local limit theorem. This paper represents the first step toward achieving this goal, specifically entailing the construction of a novel entity -- the target harmonic measure. This measure, together with the harmonic function, serves ..
arXiv:2410.06046v3 Announce Type: replace Abstract: For each simple Lie algebra $\mathfrak{g}$ of simply-laced type, Hernandez and Leclerc introduced a certain category $\mathcal{C}_{\mathbb{Z}}$ of finite-dimensional representations of the quantum affine algebra of $\mathfrak{g}$, as well as certain subcategories $\mathcal{C}_{\mathbb{Z}}^{\leq \xi}$ depending on a choice of height function adapted to an orientation of the Dynkin graph of....
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