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  • We investigate the relationship between the semiclassical wave packets of Hagedorn and the Hermite functions by establishing a relationship between their ladder operators. This Hagedorn--Hermite correspondence provides a unified view as well as simple proofs of some essential results on the Hagedorn wave packets. Particularly, we show that Hagedorn's ladder operators are a natural set of ladder operators obtained from the position and momentum operators using the symplectic group. This construction reveals an algebraic structure of the Hagedorn wave packets, and explains the relative simplicity of Hagedorn's parametrization compared to the rather intricate construction of the generalized squeezed states. We apply our formulation to show the existence of minimal uncertainty products for the Hagedorn wave packets, generalizing Hagedorn's one-dimensional result to multi-dimensions. The Hagedorn--Hermite correspondence also leads to an alternative derivation of the generating function for

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  • The Slope Conjecture relates the degree of the colored Jones polynomial of a knot to boundary slopes of incompressible surfaces. Our aim is to prove the Slope Conjecture for Montesinos knots, and to match parameters of a state-formula for the colored Jones polynomial of such knots with the parameters that describe their corresponding incompressible surfaces via the Hatcher-Oertel algorithm.

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  • Define a module representation to be a linear parameterisation of a collection of module homomorphisms over a ring. Generalising work of Knuth, we define duality functors indexed by the elements of the symmetric group of degree three between categories of module representations. We show that these functors have tame effects on average sizes of kernels. This provides a general framework for and a generalisation of duality phenomena previously observed in work of O'Brien and Voll and in the predecessor of the present article. We discuss applications to class numbers and conjugacy class zeta functions of $p$-groups and unipotent group schemes, respectively.

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  • The effective large-scale properties of materials with random heterogeneities on a small scale are typically determined by the method of representative volumes: A sample of the random material is chosen - the representative volume - and its effective properties are computed by the cell formula. Intuitively, for a fixed sample size it should be possible to increase the accuracy of the method by choosing a material sample which captures the statistical properties of the material particularly well: For example, for a composite material consisting of two constituents, one would select a representative volume in which the volume fraction of the constituents matches closely with their volume fraction in the overall material. Inspired by similar attempts in material science, Le Bris, Legoll, and Minvielle have designed a selection approach for representative volumes which performs remarkably well in numerical examples of linear materials with moderate contrast. In the present work, we provide

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  • We study the Constraint Satisfaction Problem CSP(A), where A is first-order definable in (Z;+,1) and contains +. We prove such problems are either in P or NP-complete.

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  • We prove that for each $\gamma \in (0,2)$, there is an exponent $d_\gamma > 2$, the "fractal dimension of $\gamma$-Liouville quantum gravity (LQG)", which describes the ball volume growth exponent for certain random planar maps in the $\gamma$-LQG universality class, the exponent for the Liouville heat kernel, and exponents for various continuum approximations of $\gamma$-LQG distances such as Liouville graph distance and Liouville first passage percolation. We also show that $d_\gamma$ is a continuous, strictly increasing function of $\gamma$ and prove upper and lower bounds for $d_\gamma$ which in some cases greatly improve on previously known bounds for the aforementioned exponents. For example, for $\gamma=\sqrt 2$ (which corresponds to spanning-tree weighted planar maps) our bounds give $3.4641 \leq d_{\sqrt 2} \leq 3.63299$ and in the limiting case we get $4.77485 \leq \lim_{\gamma\rightarrow 2^-} d_\gamma \leq 4.89898$.

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  • We study matching polynomials of uniform hypergraph and spectral radii of uniform supertrees. By comparing the matching polynomials of supertrees, we extend Li and Feng's results on grafting operations on graphs to supertrees. Using the methods of grafting operations on supertrees and comparing matching polynomials of supertrees, we determine the first $\lfloor\frac{d}{2}\rfloor+1$ largest spectral radii of $r$-uniform supertrees with size $m$ and diameter $d$. In addition, the first two smallest spectral radii of supertrees with size $m$ are determined.

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  • Balls are sequentially allocated into $n$ bins as follows: for each ball, an independent, uniformly random bin is generated. An overseer may then choose to either allocate the ball to this bin, or else the ball is allocated to a new independent uniformly random bin. The goal of the overseer is to reduce the load of the most heavily loaded bin after $\Theta(n)$ balls have been allocated. We provide an asymptotically optimal strategy yielding a maximum load of $(1+o(1))\sqrt{\frac{8\log n}{\log\log n}}$ balls.

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  • In the theory of Teichm\"uller space of Riemann surfaces, we consider the set of Riemann surfaces which are quasiconformally equivalent. For topologically finite Riemann surfaces, it is quite easy to examine if they are quasiconformally equivalent or not. On the other hand, for Riemann surfaces of topologically infinite type, the situation is rather complicated. In this paper, after constructing an example which shows the complexity of the problem, we give some geometric conditions for Riemann surfaces to be quasiconformally equivalent. Our argument enables us to obtain the universal Schottky space which contains all Schottky spaces, the deformation spaces of Schottky groups as the universal Teichm\"uller space contains all Teichm\"uller spaces.

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  • We study the restriction operator from the Bergman space of a domain in $\mathbb{C}^n$ to the Bergman space of a non-empty open subset of the domain. We relate the restriction operator to the Toeplitz operator on the Bergman space of the domain whose symbol is the characteristic function of the subset. Using the biholomorphic invariance of the spectrum of the associated Toeplitz operator, we study the restriction operator from the Bergman space of the unit disc to the Bergman space of subdomains with large symmetry groups, such as horodiscs and subdomains bounded by hypercycles. Furthermore, we prove a sharp estimate of the norm of the restriction operator in case the domain and the subdomain are balls. We also study various operator theoretic properties of the restriction operator such as compactness and essential norm estimates.

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  • We prove that $n$-dimensional smooth hypersurfaces of degree $n+1$ are superrigid. Starting with the work of Fano in 1915, the proof of this Theorem took 100 years and a dozen researchers to construct. Here I give complete proofs, aiming to use only basic knowledge of algebraic geometry and some Kodaira type vanishing theorems.

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  • A mathematical model for an elastoplastic porous continuum subject to large strains in combination with reversible damage (aging), evolving porosity, water and heat transfer is advanced. The inelastic response is modeled within the frame of plasticity for nonsimple materials. Water and heat diffuse through the continuum by a generalized Fick-Darcy law in the context of viscous Cahn-Hilliard dynamics and by Fourier law, respectively. This coupling of phenomena is paramount to the description of lithospheric faults, which experience ruptures (tectonic earthquakes) originating seismic waves and flash heating. In this regard, we combine in a thermodynamic consistent way the assumptions of having a small Green-Lagrange elastic strain and nearly isochoric plastification with the very large displacements generated by fault shearing. The model is amenable to a rigorous mathematical analysis. Existence of suitably defined weak solutions and a convergence result for Galerkin approximations is pr

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  • The double-layer potential plays an important role in solving boundary value problems for elliptic equations, and in the study of which for a certain equation, the properties of the fundamental solutions of the given equation are used. All the fundamental solutions of the generalized bi-axially symmetric Helmholtz equation were known, and only for the first one was constructed the theory of potential. Here, in this paper, we aim at constructing theory of double-layer potentials corresponding to the third fundamental solution. By using some properties of one of Appell's hypergeometric functions in two variables, we prove limiting theorems and derive integral equations concerning a denseness of double-layer potentials.

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  • This manuscript served as lecture notes for a mini-course in the 2016 Southern California Geometric Analysis Seminar Winter School. The goal is to give a quick introduction to Kahler geometry by describing the recent resolution of Tian's three influential properness conjectures in joint work with T. Darvas. These results---inspired by and analogous to work on the Yamabe problem in conformal geometry---give an analytic characterization for the existence of Kahler--Einstein metrics and other important canonical metrics in complex geometry, as well as strong borderline Sobolev type inequalities referred to as the (strong) Moser--Trudinger inequalities.

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  • Why deep neural networks (DNNs) capable of overfitting often generalize well in practice is a mystery in deep learning. Existing works indicate that this observation holds for both complicated real datasets and simple datasets of one-dimensional (1-d) functions. In this work, for general low-frequency dominant 1-d functions, we find that a DNN with common settings first quickly captures the dominant low-frequency components, and then relatively slowly captures high-frequency ones. We call this phenomenon Frequency Principle (F-Principle). F-Principle can be observed over various DNN setups of different activation functions, layer structures and training algorithms in our experiments. F-Principle can be used to understand (i) the behavior of DNN training in the information plane and (ii) why DNNs often generalize well albeit its ability of overfitting. This F-Principle potentially can provide insights into understanding the general principle underlying DNN optimization and generalizatio

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  • In this paper we study two deformation procedures for quantum groups --- namely, quantum universal enveloping algebras --- those realized as twist deformations (that modify the coalgebra structure, while keeping the algebra one), called "twisted quantum groups" (=TwQGp's), and those realized as 2-cocycle deformations (that deform the algebra structure, but save the coalgebra one), called "multiparameter quantum groups" (=MpQG's). Up to technicalities, we show that the two methods actually are equivalent, in that they eventually provide isomorphic outputs. In other words, the two notions of TwQG's and of MpQG's --- which, in Hopf algebra theoretical terms are naturally dual to each other --- actually coincide. Therefore, we conclude that there exists only one type of "multiparameter deformation" for universal enveloping algebras, that can be realized either as a TwQG or as a MpQG. In particular, the link between one realization and the other being just a (very simple, and rather explici

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  • Let $f(x)$ be a nonconstant polynomial with integer coefficients and nonzero discriminant. We study the distribution modulo primes of the set of squarefree integers $d$ such that the curve $dy^2=f(x)$ has a nontrivial rational or integral point.

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  • This work is devoted to study unconditionally energy stable and mass-conservative numerical schemes for the following repulsive-productive chemotaxis model: Find $u \geq 0$, the cell density, and $v \geq 0$, the chemical concentration, such that $$ \left\{ \begin{array} [c]{lll} \partial_t u - \Delta u - \nabla\cdot (u\nabla v)=0 \ \ \mbox{in}\ \Omega,\ t>0,\\ \partial_t v - \Delta v + v = u \ \ \mbox{in}\ \Omega,\ t>0, \end{array} \right. $$ in a bounded domain $\Omega\subseteq \mathbb{R}^d$, $d=2,3$. By using a regularization technique, we propose three fully discrete Finite Element (FE) approximations. The first one is a nonlinear approximation in the variables $(u,v)$; the second one is another nonlinear approximation obtained by introducing ${\boldsymbol\sigma}=\nabla v$ as an auxiliary variable; and the third one is a linear approximation constructed by mixing the regularization procedure with the energy quadratization technique, in which other auxiliary variables are intro

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  • Let $H_n$ be a graph on $n$ vertices and let $\overline{H_n}$ denote the complement of $H_n$. Suppose that $\Delta = \Delta(n)$ is the maximum degree of $\overline{H_n}$. We analyse three algorithms for sampling $d$-regular subgraphs ($d$-factors) of $H_n$. This is equivalent to uniformly sampling $d$-regular graphs which avoid a set $E(\overline{H_n})$ of forbidden edges. Here $d=d(n)$ is a positive integer which may depend on $n$. Two of these algorithms produce a uniformly random $d$-factor of $H_n$ in expected runtime which is linear in $n$ and low-degree polynomial in $d$ and $\Delta$. The first algorithm applies when $(d+\Delta)d\Delta = o(n)$. This improves on an earlier algorithm by the first author, which required constant $d$ and at most a linear number of edges in $\overline{H_n}$. The second algorithm applies when $H_n$ is regular and $d^2+\Delta^2 = o(n)$, adapting an approach developed by the first author together with Wormald. The third algorithm is a simplification of t

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  • Brauer and Fowler noted restrictions on the structure of a finite group G in terms of the order of the centralizer of an involution t in G. We consider variants of these themes. We first note that for an arbitrary finite group G of even order, we have |G| is less than the number of conjugacy classes of the Fitting subgroup times the order of the centralizer to the fourth power of any involution in G. This result does require the classification of the finite simple groups. The groups SL(2,q) with q even shows that the exponent 4 cannot be replaced by any exponent less than 3. We do not know at present whether the exponent 4 can be improved in general, though we note that the exponent 3 suffices for almost simple groups G. We are however able to prove that every finite group $G$ of even order contains an involution u such that [G:F(G)] is less than the cube of the order of the centralizer of u. There is a dichotomy in the proof of this fact, as it reduces to proving two residual cases: o

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  • We obtain optimal weak-type $(p, p)$ bounds for Bochner-Riesz means for the Hermite operator $H=-\Delta +|x|^2$ in ${\mathbb R}^n, n\geq 2$ and for other related operators for $1\leq p\leq 2n/(n+2)$, extending earlier results of Thangavelu and of Karadzhov.

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  • We construct a space of vector fields that are normal to differentiable curves in the plane. Its basis functions are defined via saddle point variational problems in reproducing kernel Hilbert spaces (RKHSs). First, we study the properties of these basis vector fields and show how to approximate them. Then, we employ this basis to discretise shape Newton methods and investigate the impact of this discretisation on convergence rates.

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  • This paper is concerned with the analysis of the quasi-static thermo-poroelastic model. This model is nonlinear and includes thermal effects compared to the classical quasi-static poroelastic model (also known as Biot's model). It consists of a momentum balance equation, a mass balance equation, and an energy balance equation, fully coupled and nonlinear due to a convective transport term in the energy balance equation. The aim of this article is to investigate, in the context of mixed formulations, the existence and uniqueness of a weak solution to this model problem. The primary variables in these formulations are the fluid pressure, temperature and elastic displacement as well as the Darcy flux, heat flux and total stress. The well-posedness of a linearized formulation is addressed first through the use of a Galerkin method and suitable a priori estimates. This is used next to study the well-posedness of an iterative solution procedure for the full nonlinear problem. A convergence p

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  • For many completely positive maps repeated compositions will eventually become entanglement breaking. To quantify this behaviour we develop a technique based on the Schmidt number: If a completely positive map breaks the entanglement with respect to any qubit ancilla, then applying it to part of a bipartite quantum state will result in a Schmidt number bounded away from the maximum possible value. Iterating this result puts a successively decreasing upper bound on the Schmidt number arising in this way from compositions of such a map. By applying this technique to completely positive maps in dimension three that are also completely copositive we prove the so called PPT squared conjecture in this dimension. We then give more examples of completely positive maps where our technique can be applied, e.g.~maps close to the completely depolarizing map, and maps of low rank. Finally, we study the PPT squared conjecture in more detail, establishing equivalent conjectures related to other parts

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  • We explicitly determine those sets of nonnegative integers which occur as sets of lengths in all numerical monoids.

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  • Elon Musk tweeted on Sunday that the company produced 7,000 cars last week, including 5,000 Model 3 electric sedans. "Beating a self-imposed deadline, the final car rolling off the assembly line on Sunday morning, several hours after the midnight goal set by Musk, two workers at the factory told Reuters on Sunday." CNBC reports: The 5,000th Model 3 finished final quality checks at the Fremont, California factory and was ready to go around 5 a.m. PDT (1200 GMT), one person told Reuters. It was not clear if Tesla could maintain that level of production for a longer period of time. Tesla had a goal of producing 5,000 Model 3s per week before the close of the second quarter on Saturday to demonstrate it could mass produce the battery-powered sedan.

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  • The Machine That Builds Itself: The Strengths of the Lisp Languages (2016)

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  • Longtime Slashdot reader TopSpin writes: The Sanmen 1 nuclear reactor in Zhejiang, China, has been synchronized to the power grid and is generating power. The reactor has been under construction for nine years and became the first AP1000 in the world to achieve criticality on June 21, 2018. The AP1000 design received final design certification from the U.S. Nuclear Regulatory Commission in 2005 and has a net output of 1.117 GWe. Three other AP1000 reactors are under construction in China at the Sanmen and Haiyang sites and two reactors are under construction in the U.S. at the Vogtle Electric Generating Plant in Georgia. On June 29, the Taishan 1 reactor became the first Areva Evolutionary Power Reactor (EPR) design to generate power. Four EPR reactors are under construction in Finland, France, and China.

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  • schwit1 shares a report: The National Security Agency has purged hundreds of millions of records logging phone calls and texts that it had gathered from American telecommunications companies since 2015, the agency has disclosed. It had realized that its database was contaminated with some files the agency had no authority to receive. The agency began destroying the records on May 23, it said in a statement. Officials had discovered "technical irregularities" this year in its collection from phone companies of so-called call record details, or metadata showing who called or texted whom and when, but not what they said. The agency had collected the data from a system it created under the USA Freedom Act. Congress enacted that law in 2015 to end and replace a once-secret program that had systematically collected Americans' domestic calling records in bulk. The National Security Agency uses the data to analyze social links between people in a hunt for hidden associates of known terrorism s

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  • Rethinking Static Reference Tables in GHC

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    SUSE acquired by EQT

    07-02 LWN 43

    SUSE has announced that it has been acquired again — this time by an investment company called EQT. Some more bright-future talk can be found in this blog entry: "In keeping with our 25-year history, SUSE intends to remain committed to an open source development and business model and actively participate in communities and projects to bring open source innovation to the enterprise as high quality, reliable and usable solutions. Our truly open, open source model, where open refers to the freedom of choice provided to customers and not just the code used in our solutions, is embedded in SUSE culture, differentiates us in the market place and has been key to our years of success."

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  •  

    Security updates for Monday

    07-02 LWN 37

    Security updates have been issued by Debian (chromium-browser, mosquitto, python-pysaml2, simplesamlphp, tiff, and tomcat7), Fedora (kernel, libgxps, nodejs, and phpMyAdmin), Mageia (ansible, firefox, java-1.8.0-openjdk, libcrypt, libgcrypt, ncurses, phpmyadmin, taglib, and webkit2), openSUSE (GraphicsMagick, ImageMagick, mailman, Opera, and rubygem-sprockets), and SUSE (ImageMagick, kernel, mariadb, and python-paramiko).

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  • Medical data tends to be messy and hard to annotate, which makes it hard for neural networks to learn from. But machine learning is beginning to make progress.

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  • Tesla stock rises after company says it met production goals, including 5,000 Model 3s.

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  • The advantages of an email-driven Git workflow

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    Unikraft Unikernel Project

    07-02 Hacker News 42

    Unikraft Unikernel Project

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    Wimbledon 2018: Andy Murray Agonistes

    07-02 THE NEW YORKER 38

    Gerald Marzorati on Andy Murray deciding not to play at Wimbledon this year, after undergoing arthroscopic surgery on his right hip, in January.

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  • With more students boasting flashy GPAs, academic honors lose their luster

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  • But, privacy experts worry about concentrating location data in one firm's hands.

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  • We show an iterated function of which iterates oscillate wildly and grow at a dizzying pace. The orbit of arbitrary positive integer always returns to 1, as in the case of Collatz function.

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  • We prove that if $S$ is an $E$-solid locally inverse semigroup, and $\rho$ is an inverse semigroup congruence on $S$ such that the idempotent classes of $\rho$ are completely simple semigroups then $S$ is embeddable into a $\lambda$-semidirect product of a completely simple semigroup by $S/\rho$. Consequently, the $E$-solid locally inverse semigroups turn out to be, up to isomorphism, the regular subsemigroups of $\lambda$-semidirect products of completely simple semigroups by inverse semigroups.

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  • Given a compact Polish space $E$ and the hyperspace of its compact subsets $\mathcal{K}(E)$, we consider the class of $G_{\delta}$ $\sigma$-ideals of compact subsets of $E$ that can be represented via a compact subset of $\mathcal{K}(E)$. If we extend such an ideal $I$ by considering $G_{\delta}$ (or analytic) sets that are covered by countable unions of sets in $I$, we show that the extended collection can still be represented via some compact subset of $\mathcal{K}(E)$.

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  • We investigate the Edge-Isoperimetric Problem (EIP) for sets with $n$ elements of the cubic lattice by emphasizing its relation with the emergence of the Wulff shape in the crystallization problem. Minimizers $M_n$ of the edge perimeter are shown to deviate from a corresponding cubic Wulff configuration with respect to their symmetric difference by at most $ {\rm O}(n^{3/4})$ elements. The exponent $3/4$ is optimal. This extends to the cubic lattice analogous results that have already been established for the triangular, the hexagonal, and the square lattice in two space dimensions.

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  • We develop an in-depth analysis of the $SO(4)$ Landau models on $S^3$ in the $SU(2)$ monopole background and their associated matrix geometry. The Schwinger and Dirac gauges for the $SU(2)$ monopole are introduced to provide a concrete coordinate representation of $SO(4)$ operators and wavefunctions. The gauge fixing enables us to demonstrate algebraic relations of the operators and the $SO(4)$ covariance of the eigenfunctions. With the spin connection of $S^3$, we construct a $SO(4)$ invariant Weyl-Landau operator and analyze its eigenvalue problem with explicit form of the eigenstates. The obtained results include the known formulae of the free Weyl operator eigenstates in the free field limit. %A synthetic connection of spin and gauge connections plays a crucial role in solving the eigenvalue problem of the relativistic Landau models. Other eigenvalue problems of variant relativistic Landau models, such as massive Dirac-Landau and supersymmetric Landau models, are investigated too.

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  • $\epsilon$-Minimum Storage Regenerating ($\epsilon$-MSR) codes form a special class of Maximum Distance Separable (MDS) codes, providing mechanisms for exact regeneration of a single code block in their codewords by downloading slighly sub-optimal amount of information from the remaining code blocks. The key advantage of these codes is a significantly lower sub-packetization that grows only logarithmically with the length of the code, while providing optimality in storage and error-correcting capacity. However, from an implementation point of view, these codes require each remaining code block to be available for the repair of any single code block. In this paper, we address this issue by constructing $\epsilon$-MSR codes that can repair a failed code block by contacting a fewer number of available code blocks. When a code block fails, our repair procedure needs to contact a few compulsory code blocks and is free to choose any subset of available code blocks for the remaining choices.

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  • We evaluate a curious determinant, first mentioned by George Andrews in 1980 in the context of descending plane partitions. Our strategy is to combine the famous Desnanot-Jacobi-Dodgson identity with automated proof techniques. More precisely, we follow the holonomic ansatz that was proposed by Doron Zeilberger in 2007. We derive a compact and nice formula for Andrews's determinant, and use it to solve a challenge problem that we posed in a previous paper. By noting that Andrews's determinant is a special case of a two-parameter family of determinants, we find closed forms for several one-parameter subfamilies. The interest in these determinants arises because they count cyclically symmetric rhombus tilings of a hexagon with several triangular holes inside.

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  • This paper studies the stochastic optimal control problem for systems with unknown dynamics. A novel decoupled data based control (D2C) approach is proposed, which solves the problem in a decoupled "open loop-closed loop" fashion that is shown to be near-optimal. First, an open-loop deterministic trajectory optimization problem is solved using a black-box simulation model of the dynamical system using a standard nonlinear programming (NLP) solver. Then a Linear Quadratic Regulator (LQR) controller is designed for the nominal trajectory-dependent linearized system which is learned using input-output experimental data. Computational examples are used to illustrate the performance of the proposed approach with three benchmark problems.

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  • Standard numerical integrators suffer from an order reduction when applied to nonlinear Schr\"{o}dinger equations with low-regularity initial data. For example, standard Strang splitting requires the boundedness of the solution in $H^{r+4}$ in order to be second-order convergent in $H^r$, i.e., it requires the boundedness of four additional derivatives of the solution. We present a new type of integrator that is based on the variation-of-constants formula and makes use of certain resonance based approximations in Fourier space. The latter can be efficiently evaluated by fast Fourier methods. For second-order convergence, the new integrator requires two additional derivatives of the solution in one space dimension, and three derivatives in higher space dimensions. Numerical examples illustrating our convergence results are included. These examples demonstrate the clear advantage of the Fourier integrator over standard Strang splitting for initial data with low regularity.

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  • The paper presents a general analytic framework to model transit systems that provide door-to-door service. The model includes as special cases non-shared taxi and demand responsive transportation (DRT). In the latter we include both, paratransit services such as dial-a-ride (DAR), and the form of ridesharing (shared taxi) currently being used by crowd-sourced taxi companies like Lyft and Uber. The framework yields somewhat optimistic results because, among other things, it is deterministic and does not track vehicles across space. By virtue of its simplicity however, the framework yields approximate closed form formulas for many cases of interest.

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  • This paper presents a rigorous treatment of the concept of extensivity in equilibrium thermodynamics from a geometric point of view. This is achieved by endowing the manifold of equilibrium states of a system with a smooth atlas that is compatible with the pseudogroup of transformations on a vector space that preserve the radial vector field. The resulting geometric structure allows for accurate definitions of extensive differential forms and scaling, and the well-known relationship between both is reproduced. This structure is represented by a global vector field that is locally written as a radial one. The submanifolds that are transversal to it are embedded, and locally defined by functions with extensive differential. These submanifolds are a geometric generalization of the space of states of a closed system in equilibrium.

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  • Given a smooth foliation by complex curves (locally around a point $x\in\mathbb{C}^2\setminus\{0\}$) which is "compatible" with the foliation by spheres centered at the origin, we construct a smooth real-valued function $g$ in a neighborhood of said point, which is positive, homogeneous and constant along the leaves. A corollary we obtain from this is relevant to the problem of "bumping out" certain pseudoconvex domains in $\mathbb{C}^3$.

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  • Recent work linking deep neural networks and dynamical systems opened up new avenues to analyze deep learning. In particular, it is observed that new insights can be obtained by recasting deep learning as an optimal control problem on difference or differential equations. However, the mathematical aspects of such a formulation have not been systematically explored. This paper introduces the mathematical formulation of the population risk minimization problem in deep learning as a mean-field optimal control problem. Mirroring the development of classical optimal control, we state and prove optimality conditions of both the Hamilton-Jacobi-Bellman type and the Pontryagin type. These mean-field results reflect the probabilistic nature of the learning problem. In addition, by appealing to the mean-field Pontryagin's maximum principle, we establish some quantitative relationships between population and empirical learning problems. This serves to establish a mathematical foundation for inves

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  • Ambiguities have recently been found in the definition of the partial derivative (in the case of presence of both explicit and implicit dependencies of the function subjected to differentiation). We investigate the possible influence of this subject on quantum mechanics and the classical/quantum field theory. Surprisingly, some commutators of operators of space-time 4-coordinates and those of 4-momenta are not equal to zero. Notes added in the Abstract: Two iterated limits are not equal each other, in general. Thus, we present an example when the massless limit of the function of E, p, m does not exist in some calculations within quantum field theory.

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  • We use analytic combinatorics to give a direct proof of the closed formula for the generating function of $p$-Bernoulli numbers.

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  • We study the behaviour of D-cap-modules on rigid analytic varieties under pushforward along a proper morphism. We prove a D-cap-module analogue of Kiehl's Proper Mapping Theorem, considering the derived sheaf-theoretic pushforward from $\mathcal{D}_X$-cap-modules to $f_*\mathcal{D}_X$-cap-modules for proper morphisms $f: X\to Y$. Under assumptions which can be naturally interpreted as a certain properness condition on the cotangent bundle, we show that any coadmissible $\mathcal{D}_X$-cap-module has coadmissible higher direct images. This implies among other things a purely geometric justification of the fact that the global sections functor in the rigid analytic Beilinson--Bernstein correspondence preserves coadmissibility, and we are able to extend this result to twisted D-cap-modules on analytified partial flag varieties.

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  • In this paper we introduce a method for solving linear and nonlinear scattering problems for wave equations using a new hybrid approach. This new approach consists of a reformulation of the governing equations into a form that can be solved by a combination of a domain-based method and a boundary-integral method.Our reformulation is aimed at a situation where we have a collection of compact scattering objects located in an otherwise homogeneous unbounded space. The domain-based method is used to propagate the equations governing the wave field inside the scattering objects forward in time. The boundary integral method is used to supply the domain-based method with the required boundary values for the wave field. In this way the best features of both methods come into play; the response inside the scattering objects, which can be caused by both material inhomogeneity and nonlinearities, is easily taken into account using the domain-based method, and the boundary conditions supplied by t

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  • We explicitly describe the solution of the G$_2$-Laplacian flow starting from an extremally Ricci-pinched closed G$_2$-structure and we investigate its properties. In particular, we show that the solution exists for all real times and that it remains extremally Ricci-pinched. We also discuss various examples.

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  • Let $b$ be a pseudo-Anosov braid whose permutation has a fixed point and let $M_b$ be the mapping torus by the pseudo-Anosov homeomorphism defined on the genus $0$ fiber $F_b$ associated with $b$. This paper describes a structure of the fibered cone $\mathcal{C}$ of $F$ for $M_b$. We prove that there is a $2$-dimensional subcone $\mathcal{C}_0$ contained in the fibered cone $ \mathcal{C}$ of $F_b$ such that the fiber $F_a$ for each primitive integral class $a \in \mathcal{C}_0$ has genus $0$. We also give a constructive description of the monodromy $ \phi_a: F_a \rightarrow F_a$ of the fibration on $M_b$ over the circle, and consequently provide a construction of many sequences of pseudo-Anosov braids with small normalized entropies. As an application we prove that the smallest entropy among skew-palindromic braids with $n$ strands is comparable to $1/n$, and the smallest entropy among elements of the odd/even spin mapping class groups of genus $g$ is comparable to $1/g$.

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  • In this paper, we develop a new nonconvex approach to the problem of low-rank and sparse matrix decomposition. In our nonconvex method, we replace the rank function and the $\ell_{0}$-norm of a given matrix with a non-convex fraction function on the singular values and the elements of the matrix respectively. An alternative direction method of multipliers algorithm is utilized to solve our nonconvex problem with the non-convex fraction function penalty. Numerical experiments on video surveillance show that our method performs very well in separating the moving objects from the static background.

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  • Solecki has shown that a broad natural class of $G_{\delta}$ ideals of compact sets can be represented through the ideal of nowhere dense subsets of a closed subset of the hyperspace of compact sets. In this note we show that the closed subset in this representation can be taken to be closed upwards.

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  • We prove two results about $\text{SLF}(\bar U_q)$, the algebra of symmetric linear forms on the restricted quantum group $\bar U_q = \bar U_q(\mathfrak{sl}(2))$. First, we express any trace on finite dimensional projective $\bar U_q$-modules as a linear combination in the basis of $\text{SLF}(\bar U_q)$ constructed by Gainutdinov - Tipunin and also by Arike. In particular, this allows us to determine the symmetric linear form corresponding to the modified trace on projective $\bar U_q$-modules. Second, we give the explicit multiplication rules between symmetric linear forms in this basis.

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  • In this note, we give a very simple description of the generalized Hamming weights of Reed--Muller codes. For this purpose, we generalize the well-known Macaulay representation of a nonnegative integer and state some of its basic properties.

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  • In the past 100 years, the research of Riemann Hypothesis meets many difficulties. Such situation may be caused by that people used to study Zeta function only regarding it as a complex function. Generally, complex functions are far more complex than real functions, and are hard to graph. So, people cannot grasp the nature of them easily. Therefore, it may be a promising way to try to correspond Zeta function to real function so that we can return to the real domain to study RH. In fact, under Laplace transform, the whole picture of Zeta function is very clear and simple, and the problem can be greatly simplified. And by Laplace transform, most integral and convolution operations can be converted into algebraic operations, which greatly simplifies calculating and analysis.

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  • The finite cell method (FCM) belongs to the class of immersed boundary methods, and combines the fictitious domain approach with high-order approximation, adaptive integration and weak imposition of unfitted Dirichlet boundary conditions. For the analysis of complex geometries, it circumvents expensive and potentially error-prone meshing procedures, while maintaining high rates of convergence. The present contribution provides an overview of recent accomplishments in the FCM with applications in structural mechanics. First, we review the basic components of the technology using the p- and B-spline versions of the FCM. Second, we illustrate the typical solution behavior for linear elasticity in 1D. Third, we show that it is straightforward to extend the FCM to nonlinear elasticity. We also outline that the FCM can be extended to applications beyond structural mechanics, such as transport processes in porous media. Finally, we demonstrate the benefits of the FCM with two application exam

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  • As a major step in their proof of Wagner's conjecture, Robertson and Seymour showed that every graph not containing a fixed graph $H$ as a minor has a tree-decomposition in which each torso is almost embeddable in a surface of bounded genus. Recently, Grohe and Marx proved a similar result for graphs not containing $H$ as a topological minor. They showed that every graph which does not contain $H$ as a topological minor has a tree-decomposition in which every torso is either almost embeddable in a surface of bounded genus, or has a bounded number of vertices of high degree. We give a short proof of the theorem of Grohe and Marx, improving their bounds on a number of the parameters involved.

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  • Consider an Archimedean partially ordered vector space $X$ with generating cone (or, more generally, a pre-Riesz space $X$). Let $P$ be a linear projection on $X$ such that both $P$ and its complementary projection $I - P$ are positive; we prove that the range of $P$ is a band. This shows that the well-known concept of band projections on vector lattices can, to a certain extent, be transferred to the framework of ordered vector spaces.

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  • We present a new and direct proof of Grothendieck's generic freeness lemma in its general form. Unlike the previously published proofs, it does not proceed in a series of reduction steps and is fully constructive, not using the axiom of choice or even the law of excluded middle. It was found by unwinding the result of a general topos-theoretic technique.

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  • This paper investigates the following quasilinear Keller-Segel-Navier-Stokes system $$ \left\{ \begin{array}{l} n_t+u\cdot\nabla n=\Delta n^m-\nabla\cdot(n\nabla c),\quad x\in \Omega, t>0, c_t+u\cdot\nabla c=\Delta c-c+n,\quad x\in \Omega, t>0,\\ u_t+\nabla P=\Delta u+n\nabla \phi,\quad x\in \Omega, t>0,\\ \nabla\cdot u=0,\quad x\in \Omega, t>0 \end{array}\right.\eqno(KSF) $$ under homogeneous boundary conditions of Neumann type for $n$ and $c$, and of Dirichlet type for $u$ in a three-dimensional bounded domains $\Omega\subseteq \mathbb{R}^3$ with smooth boundary, where %$\kappa\in \mathbb{R}$ is given %constant, $\phi\in W^{1,\infty}(\Omega),m>0$. %where $\phi\in W^{1,\infty}(\Omega),a \geq 0$ and $b > 0$. Here $g \in C^1(\bar{\Omega}\times[0,\infty))\cap L^\infty(\Omega\times(0,\infty))$, %$|S(x,n,c)|\leq(1+n)^{-\alpha}$ and the parameter $\alpha\geq0$. %For any small $\mu>0$, It is proved that if $m>\frac{4}{3}$, %$\alpha\geq\frac{5}{4}$ or %$m>\max\{\fra

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  • In \cite{listnoC3adjC4}), Borodin and Ivanova proved that every planar graph without $4$-cycles adjacent to $3$-cycle is list vertex-$2$-aborable. In fact, they proved a more general result. Inspired by these results and DP-coloring which becomes a widely studied topic, we introduce a generalization on variable degeneracy including list vertex arboricity. We use this notion to extend a general result by Borodin and Ivanova. Not only that this theorem implies results about planar graphs without $4$-cycles adjacent to $3$-cycle by Borodin and Ivanova, it implies many other results including a result by Kim and Yu \cite{KimY} that every planar graph without $4$-cycles adjacent to $3$-cycle is DP-$4$-colorable.

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  • There is a natural analogue of weak Bruhat order on the involutions in any Coxeter group, which was first considered by Richardson and Springer in the context of symmetric varieties. The saturated chains in this order from the identity to a given involution are in bijection with the reduced words for a certain set of group elements which we call atoms. We study the combinatorics of atoms for involutions in the group of signed permutations. This builds on prior work concerning atoms for involutions in the symmetric group, which was motivated by connections to the geometry of certain spherical varieties. We prove that the set of atoms for any signed involution naturally has the structure of a graded poset whose maximal elements are counted by Catalan numbers. We also characterize the signed involutions with exactly one atom and prove some enumerative results about reduced words for signed permutations.

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  • One of the simplest model of immune surveillance and neoplasia was proposed by Delisi and Resigno. Later Liu et al proved the existence of non-degenerate Takens-Bogdanov bifurcations defining a surface in the whole set of five positive parameters. In this paper we prove the existence of Bautin bifurcations completing the scenario of possible codimension two bifurcations that occur in this model. We give an interpretation of our results in terms of the three phases immunoediting theory:elimination, equilibrium and escape.

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  • In this note, we study the mapping properties of global pseudo-differential operators with symbols in Ruzhansky-Turunen classes on Besov spaces $B^{s}_{\infty,\infty}(G).$ The considered classes satisfy Fefferman type conditions of limited regularity.

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  • We consider boundary element methods where the Calder\'on projector is used for the system matrix and boundary conditions are weakly imposed using a particular variational boundary operator designed using techniques from augmented Lagrangian methods. Regardless of the boundary conditions, both the primal trace variable and the flux are approximated. We focus on the imposition of Dirichlet, mixed Dirichlet--Neumann, and Robin conditions. A salient feature of the Robin condition is that the conditioning of the system is robust also for stiff boundary conditions. The theory is illustrated by a series of numerical examples.

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  • Given a complex manifold $X$, any K\"ahler class defines an affine bundle over $X$, and any K\"ahler form in the given class defines a totally real embedding of $X$ into this affine bundle. We formulate conditions under which the affine bundles arising this way are Stein and relate this question to other natural positivity conditions on the tangent bundle of $X$. For compact K\"ahler manifolds of non-negative holomorphic bisectional curvature, we establish a close relation of this construction to adapted complex structures in the sense of Lempert--Sz\H{o}ke and to the existence question for good complexifications in the sense of Totaro. Moreover, we study projective manifolds for which the induced affine bundle is not just Stein but affine and prove that these must have big tangent bundle. In the course of our investigation, we also obtain a simpler proof of a result of Yang on manifolds having non-negative holomorphic bisectional curvature and big tangent bundle.

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  • By introducing a key combinatorial structure for words produced by a Variable Length Markov Chain (VLMC), the longest internal suffix, precise characterizations of existence and uniqueness of a stationary probability measure for a VLMC chain are given. These characterizations turn into necessary and sufficient conditions for VLMC associated to a subclass of probabilised context trees: the shift-stable context trees. As a by-product, we prove that a VLMC chain whose stabilized context tree is again a context tree has at most one stationary probability measure.

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  • Partition functions can be calculated by saddle point approximations whenever regular solutions exist, as they dominate the path integral. We examine a regular Taub-Bolt dyon of Einstein-Born-Infeld theory which has electric and magnetic flux proportional to each other. Magnetic flux is found to be essentially a winding number and the electric flux turns out to be indexed by it.

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  • In this paper, we show that steady or shrinking complete gradient Yamabe solitons with finite total scalar curvature and non-positive Ricci curvature are Ricci flat. Moreover, under certain pinching condition for Ricci curvature, we show that steady or shrinking complete gradient Yamabe solitons with finite total scalar curvature and non-positive scalar curvature have zero scalar curvature.

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  • The emergence of nonequilibrium phenomena in individual complex wave systems has long been of fundamental interests. Its analytic studies remain notoriously difficult. Using the mathematical tool of the concentration of measure (CM), we develop a theory for structures and fluctuations of waves in individual disordered media. We find that, for both diffusive and localized waves, fluctuations associated with the change in incoming waves ("wave-to-wave" fluctuations) exhibit a new kind of universalities, which does not exist in conventional mesoscopic fluctuations associated with the change in disorder realizations ("sample-to-sample" fluctuations), and originate from the coherence between the natural channels of waves -- the transmission eigenchannels. Using the results obtained for wave-to-wave fluctuations, we find the criterion for almost all stationary scattering states to exhibit the same spatial structure such as the diffusive steady state. We further show that the expectations of

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  • We consider a class of stratified groups with a CR structure and a control distance that is compatible with the CR structure. For these Lie groups we show that the space of conformal maps coincide with the space of CR and anti-CR diffeomorphisms. Furthermore, we prove that on products of such groups, all CR and anti-CR maps are product maps, up to a permutation isomorphism, and affine in each component.

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  • We propose dynamical optimal transport (OT) problems constrained in a parameterized probability subset. In application problems such as deep learning, the probability distribution is often generated by a parameterized mapping function. In this case, we derive a simple formulation for the constrained dynamical OT. We illustrate several examples of the proposed problems.

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