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Posterior distributions for community structure in sparse planted bi-section models are shown to achieve exact (resp. almost-exact) recovery, with sharp bounds for the sparsity regimes where edge probabilities decrease as $O(\log(n)/n)$ (resp. $O(1/n)$). Assuming posterior recovery, one may interpret credible sets (resp. enlarged credible sets) as asymptotically consistent confidence sets; the diameters of those credible sets are controlled by the rate of posterior concentration. If credible levels are chosen to grow to one quickly enough, corresponding credible sets can be interpreted as frequentist confidence sets without conditions on posterior concentration. In the regimes with $O(1/n)$ edge sparsity, or when within-community and between-community edge probabilities are very close, credible sets may be enlarged to achieve frequentist asymptotic coverage, also without conditions on posterior concentration., 29 pp., 3 fig

Let $R$ be a ring, and consider a left $R$-module given with two (generally infinite) direct sum decompositions, $A\oplus(\bigoplus_{i\in I} C_i)=M=B\oplus(\bigoplus_{j\in J} D_j),$ such that the submodules $A$ and $B$ and the $D_j$ are each finitely generated. We show that there then exist finite subsets $I_0\subseteq I,$ $J_0\subseteq J,$ and a direct summand $Y\subseteq \bigoplus_{i\in I_0} C_i,$ such that $A \oplus Y \ =\ B \oplus(\bigoplus_{j\in J_0} D_j).$ We then note some ways that this result can and cannot be generalized, and some related questions., Comment: 4 pages. Copy at http://math.berkeley.edu/~gbergman/papers may be updated more frequently than arXiv copy. Revision has simpler version of formula (1.2), and other, minor, improvements in wording etc

Round robin tournaments are omnipresent in sport competitions and beyond. We propose two new integer programming formulations for scheduling a round robin tournament, one of which we call the matching formulation. We analytically compare their linear relaxations with the linear relaxation of a well-known traditional formulation. We find that the matching formulation is stronger than the other formulations, while its LP relaxation is still being solvable in polynomial time. In addition, we provide an exponentially sized class of valid inequalities for the matching formulation. Complementing our theoretical assessment of the strength of the different formulations, we also experimentally show that the matching formulation is superior on a broad set of instances. Finally, we describe a branch-and-price algorithm for finding round robin tournaments that is based on the matching formulation.

Motivated by results on the rationality of equivariant Hilbert series of some hierarchical models in algebraic statistics we introduce the Segre product of formal languages and apply it to establish rationality of equivariant Hilbert series in new cases. To this end we show that the Segre product of two regular languages is again regular. We also prove that every filtration of algebras given as a tensor product of families of algebras with rational equivariant Hilbert series has a rational equivariant Hilbert series. The term equivariant is used broadly to include the action of the monoid of nonnegative integers by shifting variables. Furthermore, we exhibit a filtration of shift invariant monomial algebras that has a rational equivariant Hilbert series, but whose presentation ideals do not stabilize., 26 pages, comments are welcome!

Let $G$ be a connected graph. The eccentricity of a path $P$, denoted by ecc$_G(P)$, is the maximum distance from $P$ to any vertex in $G$. In the \textsc{Central path} (CP) problem our aim is to find a path of minimum eccentricity. This problem was introduced by Cockayne et al., in 1981, in the study of different centrality measures on graphs. They showed that CP can be solved in linear time in trees, but it is known to be NP-hard in many classes of graphs such as chordal bipartite graphs, planar 3-connected graphs, split graphs, etc. We investigate the path eccentricity of a connected graph~$G$ as a parameter. Let pe$(G)$ denote the value of ecc$_G(P)$ for a central path $P$ of $G$. We obtain tight upper bounds for pe$(G)$ in some graph classes. We show that pe$(G) \leq 1$ on biconvex graphs and that pe$(G) \leq 2$ on bipartite convex graphs. Moreover, we design algorithms that find such a path in linear time. On the other hand, by investigating the longest paths of a graph, we obtain tight upper bounds for pe$(G)$ on general graphs and $k$-connected graphs. Finally, we study the relation between a central path and a longest path in a graph. We show that on trees, and bipartite permutation graphs, a longest path is also a central path. Furthermore, for superclasses of these graphs, we exhibit counterexamples for this property.

\emph{Proto-exact categories}, introduced by Dyckerhoff and Kapranov, are a generalization of Quillen exact categories which provide a framework for defining algebraic K-theory and Hall algebras in a \emph{non-additive} setting. This formalism is well-suited to the study of categories whose objects have strong combinatorial flavor. In this paper, we show that the categories of modules over semirings and hyperrings - algebraic structures which have gained prominence in tropical geometry - carry proto-exact structures. In the first part, we prove that the category of modules over a semiring is equipped with a proto-exact structure; modules over an idempotent semiring have a strong connection to matroids. We also prove that the category of algebraic lattices $\mathcal{L}$ has a proto-exact structure, and furthermore that the subcategory of $\mathcal{L}$ consisting of finite lattices is equivalent to the category of finite $\mathbb{B}$-modules as proto-exact categories, where $\mathbb{B}$ is the \emph{Boolean semifield}. We also discuss some relations between $\mathcal{L}$ and geometric lattices (simple matroids) from this perspective. In the second part, we prove that the category of modules over a hyperring has a proto-exact structure. In the case of finite modules over the \emph{Krasner hyperfield} $\mathbb{K}$, a well-known relation between finite $\mathbb{K}$-modules and finite incidence geometries yields a combinatorial interpretation of exact sequences.

Let $p$ be a polynomial in non-commutative variables $x_1,x_2,\ldots,x_n$ with constant term zero over an algebraically closed field $K$. The object of study in this paper is the image of this kind of polynomial over the algebra of upper triangular matrices $T_m(K)$. We introduce a family of polynomials called multi-index $p$-inductive polynomials for a given polynomial $p$. Using this family we will show that, if $p$ is a polynomial identity of $T_t(K)$ but not of $T_{t+1}(K)$, then $p \left(T_m(K)\right)\subseteq T_m(K)^{(t-1)}$. Equality is achieved in the case $t=1,~m-1$ and an example has been provided to show that equality does not hold in general. We further prove existence of $d$ such that each element of $T_m(K)^{(t-1)}$ can be written as sum of $d$ many elements of $p\left( T_m(K) \right)$. It has also been shown that the image of $T_m(K)^\times$ under a word map is Zariski dense in $T_m(K)^\times$., Comment: 39 pages, 6 figures, and comments are welcome; Waring type problem is modified

In 1976 S. Eilenberg and M.-P. Schützenberger posed the following diabolical question: if $\mathbf{A}$ is a finite algebraic structure, $Σ$ is the set of all identities true in $\mathbf{A}$, and there exists a finite subset $F$ of $Σ$ such that $F$ and $Σ$ have exactly the same finite models, must there also exist a finite subset $F'$ of $Σ$ such that $F'$ and $Σ$ have exactly the same finite and infinite models? (That is, must the identities of $\mathbf{A}$ be "finitely based"?) It is known that any counter-example to their question (if one exists) must fail to be finitely based in a particularly strange way. In this paper we show that the "inherently nonfinitely based" algebras constructed by Lawrence and Willard from group actions do not fail to be finitely based in this particularly strange way, and so do not provide a counter-example to the question of Eilenberg and Schützenberger. As a corollary, we give the first known examples of inherently nonfinitely based "automatic algebras" constructed from group actions., 15 pages

Set-theoretic solutions to the Yang-Baxter equation have been studied extensively by means of related algebraic systems such as cycle sets and braces, dynamical versions of which have also been developed. No work focuses on set-theoretic solutions to the Hom-Yang-Baxter equation (HYBE for short). This paper investigates set-theoretic solutions to HYBE and associated algebraic system, called Hom-cycle sets. We characterize left non-degenerate involutive set-theoretic solutions to HYBE and Hom-cycle sets, and establish their relations. We discuss connections among Hom-cycle sets, cycle sets, left non-degenerate involutive set-theoretic solutions to HYBE and the Yang-Baxter equation.

This paper considers the problem of finding near-optimal Markovian randomized (MR) policies for finite-state-action, infinite-horizon, constrained risk-sensitive Markov decision processes (CRSMDPs). Constraints are in the form of standard expected discounted cost functions as well as expected risk-sensitive discounted cost functions over finite and infinite horizons. The main contribution is to show that the problem possesses a solution if it is feasible, and to provide two methods for finding an approximate solution in the form of an ultimately stationary (US) MR policy. The latter is achieved through two approximating finite-horizon CRSMDPs which are constructed from the original CRSMDP by time-truncating the original objective and constraint cost functions, and suitably perturbing the constraint upper bounds. The first approximation gives a US policy which is $\epsilon$-optimal and feasible for the original problem, while the second approximation gives a near-optimal US policy whose violation of the original constraints is bounded above by a specified $\epsilon$. A key step in the proofs is an appropriate choice of a metric that makes the set of infinite-horizon MR policies and the feasible regions of the three CRSMDPs compact, and the objective and constraint functions continuous. A linear-programming-based formulation for solving the approximating finite-horizon CRSMDPs is also given., Comment: 38 pages

For unweighted graphs, finding isometric embeddings is closely related to decompositions of $G$ into Cartesian products of smaller graphs. When $G$ is isomorphic to a Cartesian graph product, we call the factors of this product a factorization of $G$. When $G$ is isomorphic to an isometric subgraph of a Cartesian graph product, we call those factors a pseudofactorization of $G$. Prior work has shown that an unweighted graph's pseudofactorization can be used to generate a canonical isometric embedding into a product of the smallest possible pseudofactors. However, for arbitrary weighted graphs, which represent a richer variety of metric spaces, methods for finding isometric embeddings or determining their existence remain elusive, and indeed pseudofactorization and factorization have not previously been extended to this context. In this work, we address the problem of finding the factorization and pseudofactorization of a weighted graph $G$, where $G$ satisfies the property that every edge constitutes a shortest path between its endpoints. We term such graphs minimal graphs, noting that every graph can be made minimal by removing edges not affecting its path metric. We generalize pseudofactorization and factorization to minimal graphs and develop new proof techniques that extend the previously proposed algorithms due to Graham and Winkler [Graham and Winkler, '85] and Feder [Feder, '92] for pseudofactorization and factorization of unweighted graphs. We show that any $m$-edge, $n$-vertex graph with positive integer edge weights can be factored in $O(m^2)$ time, plus the time to find all pairs shortest paths (APSP) distances in a weighted graph, resulting in an overall running time of $O(m^2+n^2\log\log n)$ time. We also show that a pseudofactorization for such a graph can be computed in $O(mn)$ time, plus the time to solve APSP, resulting in an $O(mn+n^2\log\log n)$ running time., Comment: 37 pages, 10 figures

In this paper, we derive a recurrence relation of anti-forcing polynomial for catacondensed hexagonal systems.

Let $k$ be any positive integer and $G$ a compact (Hausdorff) group. Let $\mf{np}_k(G)$ denote the probability that $k+1$ randomly chosen elements $x_1,\dots,x_{k+1}$ satisfy $[x_1,x_2,\dots,x_{k+1}]=1$. We study the following problem: If $\mf{np}_k(G)>0$ then, does there exist an open nilpotent subgroup of class at most $k$? The answer is positive for profinite groups and we give a new proof. We also prove that the connected component $G^0$ of $G$ is abelian and there exists a closed normal nilpotent subgroup $N$ of class at most $k$ such that $G^0N$ is open in $G$.

We introduce a variant of global generation for coherent sheaves on abelian varieties which, under certain circumstances, implies ampleness. This extends a criterion of Debarre asserting that a continuously globally generated coherent sheaf on an abelian variety is ample. We apply this to show the ampleness of certain sheaves, which we call naive Fourier-Mukai-Poincar\'e transforms, and to study the structure of GV sheaves. In turn, one of these applications allows to extend the classical existence and connectedness results of Brill-Noether theory to a wider context, e.g. singular curves equipped with a suitable morphism to an abelian variety. Another application is a general inequality of Brill-Noether type involving the Euler characteristic and the homological dimension., Comment: 23 pages. Proposition 3.1.3 added. Other minor changes

This thesis presents work aimed at improving the accuracy of sequence-based signal peptide prediction methods by incorporating structural information. Although sequence-based methods have become increasingly accurate, even state-of-the-art predictors commonly produce false positive signal peptide predictions. Since the presence or absence of a signal peptide is crucial in determining a protein’s subcellular localization, accuracy in identifying these signals is essential. My work attempts to improve the accuracy of subcellular localization by building structure-based approaches to refine signal peptide predictions. The thesis first describes the development of a refinement method that specifically focuses on signal peptides predicted in the Escherichia coli proteome. The refinement approach follows from the observation that the region corresponding to an incorrectly predicted signal peptide generally associates with the body of the protein in the protein structure. Predicted structural information from the trRosetta web-server was incorporated in the form of inter-residue distance (contact) maps, which could be used to gauge the extent of interaction between potential signal peptides and the protein body. The classification model built based on trRosetta contact maps proved to be highly accurate in correctly classifying true false positive predicted signal peptides., Focusing next on the extension of the approach to other organisms, the thesis presents a more general approach for improving signal peptide prediction methods that takes advantage of the highly accurate protein structures predicted by AlphaFold2. It targets the most recent signal peptide prediction methods for both the prokaryotes and eukaryotes. Eventually, the method was applied to probe false positive signal peptides predicted from the proteomes of forty-eight organisms whose protein structures are directly available in the AlpahFold2 structural database., Finally, the thesis discusses the application of predicted structures of protein complexes in determining the specificity of enzymes towards predicted signal peptides. This chapter highlights several applications of the method, including the prediction of cleavage sites as well as the determination of enzyme specificity. Overall, the study demonstrates that analysis of a protein’s structure can be a powerful tool in predicting its subcellular localization.

In this paper, we propose a new spectral-based approach to hypothesis testing for populations of networks. The primary goal is to develop a test to determine whether two given samples of networks come from the same random model or distribution. Our test statistic is based on the trace of the third order for a centered and scaled adjacency matrix, which we prove converges to the standard normal distribution as the number of nodes tends to infinity. The asymptotic power guarantee of the test is also provided. The proper interplay between the number of networks and the number of nodes for each network is explored in characterizing the theoretical properties of the proposed testing statistics. Our tests are applicable to both binary and weighted networks, operate under a very general framework where the networks are allowed to be large and sparse, and can be extended to multiple-sample testing. We provide an extensive simulation study to demonstrate the superior performance of our test over existing methods and apply our test to three real datasets.

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Selecting hyperparameters for unsupervised learning problems is challenging in general due to the lack of ground truth for validation. Despite the prevalence of this issue in statistics and machine learning, especially in clustering problems, there are not many methods for tuning these hyperparameters with theoretical guarantees. In this paper, we provide a framework with provable guarantees for selecting hyperparameters in a number of distinct models. We consider both the subgaussian mixture model and network models to serve as examples of i.i.d. and non-i.i.d. data. We demonstrate that the same framework can be used to choose the Lagrange multipliers of penalty terms in semi-definite programming (SDP) relaxations for community detection, and the bandwidth parameter for constructing kernel similarity matrices for spectral clustering. By incorporating a cross-validation procedure, we show the framework can also do consistent model selection for network models. Using a variety of simulated and real data examples, we show that our framework outperforms other widely used tuning procedures in a broad range of parameter settings.

We refer by threshold Ornstein-Uhlenbeck to a continuous-time threshold autoregressive process. It follows the Ornstein-Uhlenbeck dynamics when above or below a fixed level, yet at this level (threshold) its coefficients can be discontinuous. We discuss (quasi)-maximum likelihood estimation of the drift parameters, both assuming continuous and discrete time observations. In the ergodic case, we derive consistency and speed of convergence of these estimators in long time and high frequency. Based on these results, we develop a test for the presence of a threshold in the dynamics. Finally, we apply these statistical tools to short-term US interest rates modeling.

We assume a spatial blind source separation model in which the observed multivariate spatial data is a linear mixture of latent spatially uncorrelated random fields containing a number of pure white noise components. We propose a test on the number of white noise components and obtain the asymptotic distribution of its statistic for a general domain. We also demonstrate how computations can be facilitated in the case of gridded observation locations. Based on this test, we obtain a consistent estimator of the true dimension. Simulation studies and an environmental application in the Supplemental Material demonstrate that our test is at least comparable to and often outperforms bootstrap-based techniques, which are also introduced in this paper. peerReviewed

High-dimensional self-exciting point processes have been widely used in many application areas to model discrete event data in which past and current events affect the likelihood of future events. In this paper, we are concerned with detecting abrupt changes of the coefficient matrices in discrete-time high-dimensional self-exciting Poisson processes, which have yet to be studied in the existing literature due to both theoretical and computational challenges rooted in the non-stationary and high-dimensional nature of the underlying process. We propose a penalized dynamic programming approach which is supported by a theoretical rate analysis and numerical evidence.

We consider $n\times n$ real-valued matrices $A = (a_{ij})$ satisfying $a_{ii} \geq a_{i,i+1} \geq \dots \geq a_{in} \geq a_{i1} \geq \dots \geq a_{i,i-1}$ for $i = 1,\dots,n$. With such a matrix $A$ we associate a directed graph $G(A)$. We prove that the solutions to the system $A^T x = \lambda e$, with $\lambda \in \mathbb{R}$ and $e$ the vector of all ones, are linear combinations of 'fundamental' solutions to $A^T x=e$ and vectors in $\ker A^T$, each of which is associated with a closed strongly connected component (SCC) of $G(A)$. This allows us to characterize the sign of $\det A$ in terms of the number of closed SCCs and the solutions to $A^T x = e$. In addition, we provide conditions for $A$ to be a $P$-matrix., Comment: 17 pages, 2 figures; minor changes in introduction, added Figure 1, corrected typos

We provide a practical technique to obtain plenty of algebraic relations for theta functions on the bounded symmetric domains of type $I$. In our framework, each theta relation is controlled by combinatorial properties of a pair $(T,P)$ of a regular matrix $T$ over an imaginary quadratic field and a positive-definite Hermitian matrix $P$ over the complex number field., Comment: 16 pages, typos are corrected, unclear descriptions are corrected, abstract is revised

In this paper, we study the matrix period and the competition period of Toeplitz matrices over a binary Boolean ring $\mathbb{B} = \{0,1\}$. Given subsets $S$ and $T$ of $\{1,\ldots,n-1\}$, an $n\times n$ Toeplitz matrix $A=T_n\langle S ; T \rangle$ is defined to have $1$ as the $(i,j)$-entry if and only if $j-i \in S$ or $i-j \in T$. We show that if $\max S+\min T \le n$ and $\min S+\max T \le n$, then $A$ has the matrix period $d/d'$ and the competition period $1$ where $d = \gcd (s+t \mid s \in S, t \in T)$ and $d' = \gcd(d, \min S)$. Moreover, it is shown that the limit of the matrix sequence $\{A^m(A^T)^m\}_{m=1}^\infty$ is a directed sum of matrices of all ones except zero diagonal. In many literatures we see that graph theoretic method can be used to prove strong structural properties about matrices. Likewise, we develop our work from a graph theoretic point of view.

We lower bound the rank of a tensor by a linear combination of the ranks of three of its unfoldings, using Sylvester's rank inequality. In a similar way, we lower bound the symmetric rank by a linear combination of the symmetric ranks of three unfoldings. Lower bounds on the rank and symmetric rank of tensors are important for finding counterexamples to Comon's conjecture. A real counterexample to Comon's conjecture is a tensor whose real rank and real symmetric rank differ. Previously, only one real counterexample was known. We divide the construction into three steps. The first step involves linear spaces of binary tensors. The second step considers a linear space of larger decomposable tensors. The third step is to verify a conjecture that lower bounds the symmetric rank, on a tensor of interest. We use the construction to build an order six real tensor whose real rank and real symmetric rank differ., 26 pages, 3 figures, v2: updated to match published version

A classic result in extremal graph theory, known as Mantel's theorem, states that every non-bipartite graph of order $n$ with size $m>\lfloor \frac{n^{2}}{4}\rfloor$ contains a triangle. Lin, Ning and Wu [Comb. Probab. Comput. 30 (2021) 258-270] proved a spectral version of Mantel's theorem for given order $n.$ Zhai and Shu [Discrete Math. 345 (2022) 112630] investigated a spectral version for fixed size $m.$ In this paper, we prove $Q$-spectral versions of Mantel's theorem., 14 pages, 4 figures

It has been recently observed that fundamental aspects of the classical theory of factorization can be greatly generalized by combining the languages of monoids and preorders. This has led to various theorems on the existence of certain factorizations, herein called $\preceq$-factorizations, for the $\preceq$-non-units of a (multiplicatively written) monoid $H$ endowed with a preorder $\preceq$, where an element $u \in H$ is a $\preceq$-unit if $u \preceq 1_H \preceq u$ and a $\preceq$-non-unit otherwise. The ``building blocks'' of these factorizations are the $\preceq$-irreducibles of $H$ (i.e., the $\preceq$-non-units $a \in H$ that cannot be written as a product of two $\preceq$-non-units each of which is strictly $\preceq$-smaller than $a$); and it is interesting to look for sufficient conditions for the $\preceq$-factorizations of a $\preceq$-non-unit to be bounded in length or finite in number (if measured or counted in a suitable way). This is precisely the kind of questions addressed in the present work, whose main novelty is the study of the interaction between minimal $\preceq$-factorizations (i.e., a refinement of $\preceq$-factorizations used to counter the ``blow-up phenomena'' that are inherent to factorization in non-commutative or non-cancellative monoids) and some finiteness conditions describing the ``local behaviour'' of the pair $(H, \preceq)$. Besides a number of examples and remarks, the paper includes many arithmetic results, a part of which are new already in the basic case where $\preceq$ is the divisibility preorder on $H$ (and hence in the setup of the classical theory)., Comment: 27 pages, 2 figures. Final version to appear in Journal of Algebra

We show that the space of invariants for the Weil representation for discriminant groups which contain self-dual isotropic subgroups is spanned by the characteristic functions of the self-dual isotropic subgroups. As an application, we construct modular units for orthogonal groups in signature (2,2) using Borcherds products., Comment: 24 pages, final version, to appear in Journal of Number Theory

We introduce the notion of residual intersections of modules and prove their existence. We show that projective dimension one modules have Cohen-Macaulay residual intersections, namely they satisfy the relevant Artin-Nagata property. We then establish a formula for the core of orientable modules satisfying certain homological conditions, extending previous results of Corso, Polini, and Ulrich on the core of projective one modules. Finally, we provide examples of classes of modules that satisfy our assumptions.

We consider the linear field-road system, a model for fast diffusion channels in population dynamics and ecology. This system takes the form of a system of PDEs set on domains of different dimensions, with exchange boundary conditions. Despite the intricate geometry of the problem, we provide an explicit expression for its fundamental solution and for the solution to the associated Cauchy problem. The main tool is a Fourier (on the road variable)/Laplace (on time) transform. In addition, we derive estimates for the decay rate of the L $\infty$ norm of these solutions.

We describe an algorithm that constructs a list of all topological types of holomorphic actions of a finite group on a compact Riemann surface $C$ of genus at least $g \geq 2$ with $C/G \cong \mathbb{P}^1$., v2: corrected definition of Hurwitz equivalence using the braid group

A soluble pro-p group of finite rank is finitely axiomatizable in the class of all profinite groups if and only if for each open subgroup H, the image of Z(H) in the abelianization of H is finite, subject to some suitable hypothesis of finite presentability., Solves Problem 1 in `Finite axiomatizability for profinite groups', Proc. London Math. Soc.(3) 123, (2021)

For asymmetric sinh-Poisson type problems with Dirichlet boundary condition arising as a mean field equation of equilibrium turbulence vortices with variable intensities of interest in hydrodynamic turbulence, we address the existence of sign-changing bubble tower solutions on a pierced domain $\Omega_\epsilon:=\Omega\setminus \displaystyle \overline{B(\xi,\epsilon)}$, where $\Omega$ is a smooth bounded domain in $\mathbb{R}^2$ and $B(\xi,\epsilon)$ is a ball centered at $\xi\in \Omega$ with radius $\epsilon>0$. Precisely, given a small parameter $\rho>0$ and any integer $m\ge 2$, there exist a radius $\epsilon=\epsilon(\rho)>0$ small enough such that each sinh-Poisson type equation, either in Liouville form or mean field form, has a solution $u_\rho$ with an asymptotic profile as a sign-changing tower of $m$ singular Liouville bubbles centered at the same $\xi$ and with $\epsilon(\rho)\to 0^+$ as $\rho$ approaches to zero., Comment: arXiv admin note: text overlap with arXiv:1904.00127, arXiv:1909.00905

A Leavitt labelled path algebra over a commutative unital ring is associated with a labelled space, generalizing Leavitt path algebras associated with graphs and ultragraphs as well as torsion-free commutative algebras generated by idempotents. We show that Leavitt labelled path algebras can be realized as partial skew group rings, Steinberg algebras, and Cuntz-Pimsner algebras. Via these realizations we obtain generalized uniqueness theorems, a description of diagonal preserving isomorphisms and we characterize simplicity of Leavitt labelled path algebras. In addition, we prove that a large class of partial skew group rings can be realized as Leavitt labelled path algebras.

In this note, we prove pointwise decay in time of solutions to the 3D energy-critical nonlinear Schrödinger equations assuming data in $L^1\cap H^3$. The main ingredients are the boundness of the Schrödinger propagators in Hardy space due to Miyachi \cite{Miyachi} and a fractional Leibniz rule in the Hardy space. We also extend the fractional chain rule to the Hardy space., 9 pages; In this new version, we extend the fractional chain rule to the Hardy space

We consider a two-tensor hydrodynamics derived from the molecular model, where high-order tensors are determined by closure approximation through the maximum entropy state or the quasi-entropy. We prove the existence and uniqueness of local in time smooth solutions to the two-tensor system. Then, we rigorously justify the connection between the molecular-theory-based two-tensor hydrodynamics and the biaxial frame hydrodynamics. More specifically, in the framework of Hilbert expansion, we show the convergence of the solution to the two-tensor hydrodynamics to the solution to the frame hydrodynamics., 41 pages

The Kato's decomposition \cite[Theorem 4]{kato} is generalized to semi-B-Fredholm operators.

For a sequence of polynomials $\{p_k(t)\}$ in one real or complex variable, where $p_k$ has degree $k$, for $k\ge 0$, we find explicit expressions and recurrence relations for infinite matrices whose entries are the coefficients $d(n,m,k)$, called linearization coefficients, that satisfy $$ p_n(t) p_m(t)=\sum_{k=0}^{n+m} d(n,m,k) p_k(t).$$ For any pair of polynomial sequences $\{u_k(t)\}$ and $\{p_k(t)\}$ we find infinite matrices whose entries are the coefficients $e(n,m,k)$ that satisfy $$p_n(t) p_m(t)=\sum_{k=0}^{n+m} e(n,m,k) u_k(t).$$ Such results are obtained using a matrix approach. We also obtain recurrence relations for the linearization coefficients, apply the general results to general orthogonal polynomial sequences and to particular families of orthogonal polynomials such as the Chebyshev, Hermite, and Charlier families., 14 pages

We describe a practical algorithm for computing normal forms for semigroups and monoids with finite presentations satisfying so-called small overlap conditions. Small overlap conditions are natural conditions on the relations in a presentation, which were introduced by J. H. Remmers and subsequently studied extensively by M. Kambites. Presentations satisfying these conditions are ubiquitous; Kambites showed that a randomly chosen finite presentation satisfies the $C(4)$ condition with probability tending to 1 as the sum of the lengths of relation words tends to infinity. Kambites also showed that several key problems for finitely presented semigroups and monoids are tractable in $C(4)$ monoids: the word problem is solvable in $O(\min\{|u|, |v|\})$ time in the size of the input words $u$ and $v$; the uniform word problem for $\langle A|R\rangle$ is solvable in $O(N ^ 2 \min\{|u|, |v|\})$ where $N$ is the sum of the lengths of the words in $R$; and a normal form for any given word $u$ can be found in $O(|u|)$ time. Although Kambites' algorithm for solving the word problem in $C(4)$ monoids is highly practical, it appears that the coefficients in the linear time algorithm for computing normal forms are too large in practice. In this paper, we present an algorithm for computing normal forms in $C(4)$ monoids that has time complexity $O(|u| ^ 2)$ for input word $u$, but where the coefficients are sufficiently small to allow for practical computation. Additionally, we show that the uniform word problem for small overlap monoids can be solved in $O(N \min\{|u|, |v|\})$ time., Comment: 31 pages, 6 figures (some further minor corrections from the referee are resolved, to appear in Journal of Algebra)

We introduce a family of quiver Hecke algebras which give a categorification of quantum Borcherds algebra associated to an arbitrary Borcherds-Cartan datum.

A $k$-edge-weighting of $G$ is a mapping $\omega:E(G)\longrightarrow \{1,\ldots,k\}$. The edge-weighting of $G$ naturally induces a vertex-colouring $\sigma_{\omega}:V(G)\longrightarrow \mathbb{N}$ given by$\sigma_{\omega}(v)=\sum_{u\in N_G(v)}\omega(vu)$ for every $v\in V(G)$. The edge-weighting $\omega$ is neighbour sum distinguishing if it yields a proper vertex-colouring $\sigma_{\omega}$, \emph{i.e.}, $\sigma_{\omega}(u)\neq \sigma_{\omega}(v)$ for every edge $uv$ of $G$.We investigate a neighbour sum distinguishing edge-weighting with local constraints, namely, we assume that the set of edges incident to a vertex of large degree is not monochromatic. A graph is nice if it has no components isomorphic to $K_2$. We prove that every nice graph with maximum degree at most~5 admits a neighbour sum distinguishing $(\Delta(G)+2)$-edge-weighting such that all the vertices of degree at least~2 are incident with at least two edges of different weights. Furthermore, we prove that every nice graph admits a neighbour sum distinguishing $7$-edge-weighting such that all the vertices of degree at least~6 are incident with at least two edges of different weights. Finally, we show that nice bipartite graphs admit a neighbour sum distinguishing $6$-edge-weighting such that all the vertices of degree at least~2 are incident with at least two edges of different weights.

We consider linear boundary value problems for higher-order parameter-elliptic equations, where the boundary data do not belong to the classical trace spaces. We employ a class of Sobolev spaces of mixed smoothness that admits a generalized boundary trace with values in Besov spaces of negative order. We prove unique solvability for rough boundary data in the half-space and in sufficiently smooth domains. As an application, we show that the operator related to the linearized Cahn--Hilliard equation with dynamic boundary conditions generates a holomorphic semigroup in $L^p(\mathbb R^n_+)\times L^p(\mathbb R^{n-1})$., 41 pages

We study a relation between Hadamard powers and polynomial kernel perceptrons. The rank of Hadamard powers for the special case of a Boolean matrix and for the generic case of a real matrix is computed explicitly. These results are interpreted in terms of the classification capacities of perceptrons., Comment: 12 pages

In this paper, we first recall the notion of (noncommutative) Poisson conformal algebras and describe some constructions of them. Then we study the formal distribution (noncommutative) Poisson algebras and coefficient (noncommutative) Poisson algebras. Next, we introduce the notion of conformal formal deformations of commutative associative conformal algebras and show that Poisson conformal algebras are the corresponding semi-classical limits. At last, we develop the cohomology theory of noncommutative Poisson conformal algebras and use this cohomology to study their deformations., 31 pages

A splinter is a notion of singularity that has seen numerous recent applications, especially in connection with the direct summand theorem, the mixed characteristic minimal model program, Cohen-Macaulayness of absolute integral closures and cohomology vanishing theorems. Nevertheless, many basic questions about these singularities remain elusive. One outstanding problem is whether the splinter property spreads from a point to an open neighborhood of a noetherian scheme. Our paper addresses this problem in prime characteristic, where we show that a locally noetherian scheme that has finite Frobenius or that is locally essentially of finite type over a quasi-excellent local ring has an open splinter locus. In particular, all varieties over fields of positive characteristic have open splinter loci. Intimate connections are established between the openness of splinter loci and $F$-compatible ideals, which are prime characteristic analogues of log canonical centers. We prove the surprising fact that for a large class of noetherian rings with pure (aka universally injective) Frobenius, the splinter condition is detected by the splitting of a single generically \'etale finite extension. We also show that for a noetherian $\textbf{N}$-graded ring over a field, the homogeneous maximal ideal detects the splinter property., Comment: Comments welcome, minor updates in v2

We study the asymptotic emergent dynamics of two models that can be thought of as extensions of the well known Schr\"odinger-Lohe model for quantum synchronization. More precisely, the interaction strength between different oscillators is determined by intrinsic parameters, following Cucker-Smale communication protocol. Unlike the original Schr\"odinger-Lohe system, where the interaction strength was assumed to be uniform, in the cases under our consideration the total mass of each quantum oscillator is allowed to vary in time. A striking consequence of this property is that these extended models yield configurations exhibiting phase, but not space, synchronization. The results are mainly based on the analysis of the ODE systems arising from the correlations, control over the well known Cucker-Smale dynamics, and the dynamics satisfied by the quantum order parameter., Comment: 18 pages, minor changes and submitted

The article focuses on four aspects related to the descent algebras of type $A$. They are modular idempotents, higher Lie powers, higher Lie modules and the right ideals of the symmetric group algebras generated by the Solomon's descent elements. More precisely, we give a construction for the modular idempotents, describe the dimension and character for higher Lie powers and study the structures of the higher Lie modules and the right ideals both in the ordinary and modular cases., Comment: This expands the previous version by including extra sections (4 and 6)

Saddle point problems arise in a variety of applications, e.g., when solving the Stokes equations. They can be formulated such that the system matrix is symmetric, but indefinite, so the variational convergence theory that is usually used to prove multigrid convergence cannot be applied. In a 2016 paper in Numerische Mathematik Notay has presented a different algebraic approach that analyzes properly preconditioned saddle point problems, proving convergence of the Two-Grid method. In the present paper we analyze saddle point problems where the blocks are circulant within this framework. We are able to derive sufficient conditions for convergence and provide optimal parameters for the preconditioning of the saddle point problem and for the point smoother that is used. The analysis is based on the generating symbols of the circulant blocks. Further, we show that the structure can be kept on the coarse level, allowing for a recursive application of the approach in a W- or V-cycle and proving the "level independency" property. Numerical results demonstrate the efficiency of the proposed method in the circulant and the Toeplitz case.

This paper studies limit measures of stationary measures of stochastic ordinary differential equations on the Euclidean space and tries to determine which invariant measures of an unperturbed system will survive. Under the assumption for SODEs to admit the Freidlin-Wentzell or Dembo-Zeitouni large deviations principle with weaker compactness condition, we prove that limit measures are concentrated away from repellers which are topologically transitive, or equivalent classes, or admit Lebesgue measure zero. We also preclude concentrations of limit measures on acyclic saddle or trap chains. This illustrates that limit measures are concentrated on Liapunov stable compact invariant sets. Applications are made to the Morse-Smale systems, the Axiom A systems including structural stability systems and separated star systems, the gradient or gradient-like systems, those systems possessing the Poincare-Bendixson property with a finite number of limit sets to obtain that limit measures live on Liapunov stable critical elements, Liapunov stable basic sets, Liapunov stable equilibria and Liapunov stable limit sets including equilibria, limit cycles and saddle or trap cycles, respectively. A number of nontrivial examples admitting a unique limit measure are provided, which include monostable, multistable systems and those possessing infinite equivalent classes., Comment: 37 pages, 12 figures

For $q\in\mathbb{R}$, the $Q$-matrix $Q=Q_q$ of a connected simple graph $G=(V,E)$ is $Q_q=(q^{\partial(x,y)})_{x,y\in V}$, where $\partial$ denotes the path-length distance. Describing the set $\pi(G)$ consisting of those $q\in \mathbb{R}$ for which $Q_q$ is positive semidefinite is fundamental in asymptotic spectral analysis of graphs from the viewpoint of quantum probability theory. Assume that $G$ has at least two vertices. Then $\pi(G)$ is easily seen to be a nonempty closed subset of the interval $[-1,1]$. In this note, we show that $\pi(G)=[-1,1]$ if and only if $G$ is isometrically embeddable into a hypercube (infinite-dimensional if $G$ is infinite) if and only if $G$ is bipartite and does not possess certain five-vertex configurations, an example of which is an induced $K_{2,3}$., Comment: 6 pages