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In this paper we study dynamic backward problems, with the computation of conditional expectations as a main objective, in a framework where the (forward) state process satisfies a Volterra type SDE, with fractional Brownian motion as a typical example. Such processes are neither Markov processes nor semimartingales, and most notably, they feature a certain time inconsistency which makes any direct application of Markovian ideas, such as flow properties, impossible without passing to a path-dependent framework. Our main result is a functional It\^{o} formula, extending the seminal work of Dupire \cite{Dupire} to our more general framework. In particular, unlike in \cite{Dupire} where one needs only to consider the stopped paths, here we need to concatenate the observed path up to the current time with a certain smooth observable curve derived from the distribution of the future paths. This new feature is due to the time inconsistency involved in this paper. We then derive the path depe
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Following the proof given by Froese and Herbst in [FH82] with another conjugate operator, we show for a class of real potential that possible eigenfunction of the Schr\"odinger operator has to decay sub-exponentially. We also show that, for a certain class of potential, this bound can not be satisfied which implies the absence of strictly positive eigenvalues for the Schr\"odinger operator.
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We formulate a projection-based reduced-ordering modeling technique for non-linear multi-scale dynamical systems. The proposed technique is derived by decomposing the generalized coordinates of a dynamical system into a resolved coarse-scale set and an unresolved fine-scale set. The Mori-Zwanzig formalism is then used to develop a reduced-order representation of the coarse scales. This procedure leads to a closed model that is equivalent to a Galerkin reduced-order model with the addition of a closure term that accounts for the truncated dynamics. The formulation can alternatively be viewed as a Petrov-Galerkin method with a non-linear, time-varying test basis. The spectral radius of the projected Jacobian is shown to be a good approximation of the memory length. Numerical experiments on the compressible Navier-Stokes equations in one and two-dimensions demonstrate that the proposed method leads to improvements over the standard Galerkin ROM and, in some cases, over the least-squares P
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In the present article, we investigate the following deformation problem. Let $(R,\mathfrak m)$ be a local (graded local) Noetherian ring with a (homogeneous) regular element $y \in \mathfrak m$ and assume that $R/yR$ is quasi-Gorenstein. Then is $R$ quasi-Gorenstein? We give positive answers to this problem under various assumptions, while we present a counter-example in general. We emphasize that absence of the Cohen-Macaulay condition requires some delicate studies.
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Let $F$ be a $2$-regular graph of order $v$. The Oberwolfach problem, $OP(F)$, asks for a $2$-factorization of the complete graph on $v$ vertices in which each $2$-factor is isomorphic to $F$. Posed by G. Ringel in the 1960s, this problem is still open, even though infinitely many cases have been solved. For example, complete solutions are known for infinitely many prime orders $[3]$, when $F$ consists of cycles of the same length $[12]$, or when $F$ has two components $[21]$. In this paper, we give a complete solution of the Oberwolfach problem over infinite complete graphs, proving the existence of solutions that are regular under the action of a given involution free group $G$. Moreover, we characterize the infinite subgraphs $E$ of $F$ such that there exists a solution to $OP(F)$ containing a solution to $OP(E)$.
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The main purpose of this paper is to give a generalization of Dijkgraaf-Witten theory. Consider a morphism from a smash product of spectra E,F to another spectrum G. We construct a TQFT for E-oriented manifolds using a representative of an F-cohomology class of the classifying space of a finite group. For the case that each of spectra E,F,G is given as the K-theory spectrum KU, we further generalize our construction based on non-commutative settings.
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The final purpose of this paper is to show that, by inserting a convexity constraint on the cables of a suspension bridge, the torsional instability of the deck appears at lower energy thresholds. Since this constraint is suggested by the behavior of real cables, this model appears more reliable than the classical ones. Moreover, it has the advantage to reduce to two the number of degrees of freedom (DOF), avoiding to introduce the slackening mechanism of the hangers. The drawback is that the resulting energy functional is extremely complicated, involving the convexification of unknown functions. This paper is divided in two main parts. The first part is devoted to the study of these functionals, through classical methods of calculus of variations. The second part applies this study to the suspension bridge model with convexified cables.
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Our goal is to develop spectral and scattering theories for the one-dimensional Schr\"odinger operator with a long-range potential $q(x)$, $x\geq 0$. Traditionally, this problem is studied with a help of the Green-Liouville approximation. This requires conditions on the first two derivatives $q' (x)$ and $q'' (x)$. We suggest a new Ansatz that allows us to develop a consistent theory under the only assumption $q' \in L^1$.
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A polynomial $f(x)$ over a field $K$ is called stable if all of its iterates are irreducible over $K$. In this paper we study the stability of trinomials over finite fields. Specially, we show that if $f(x)$ is a trinomial of even degree over the binary field $\mathbb{F}_2$, then $f(x)$ is not stable. We prove a similar result for some families of monic trinomials over finite fields of odd characteristic. These results are obtained towards the resolution of a conjecture on the instability of polynomials over finite fields whose degrees are divisible by the characteristic of the underlying field.
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We construct a bialgebra object in the category of linear maps LMfrom a cocommutative rack bialgebra. Theconstruction does extend to somenon-cocommutative rack bialgberas, as isillustrated by a concrete example. As a separate result, we show that the Loday complex with adjoint coefficients embeds into the rack bialgebra deformation complex for the rack bialgebra defined by a Leibniz algebra.
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6d superconformal field theories (SCFTs) are the SCFTs in the highest possible dimension. They can be geometrically engineered in F-theory by compactifying on non-compact elliptic Calabi-Yau manifolds. In this paper we focus on the class of SCFTs whose base geometry is determined by $-2$ curves intersecting according to ADE Dynkin diagrams and derive the corresponding mirror Calabi-Yau manifold. The mirror geometry is uniquely determined in terms of the mirror curve which has also an interpretation in terms of the Seiberg-Witten curve of the four-dimensional theory arising from torus compactification. Adding the affine node of the ADE quiver to the base geometry, we connect to recent results on SYZ mirror symmetry for the $A$ case and provide a physical interpretation in terms of little string theory. Our results, however, go beyond this case as our construction naturally covers the $D$ and $E$ cases as well.
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In this paper, we show that the standard semidefinite programming (SDP) relaxation of altering current optimal power flow (AC OPF) can be equivalently reformulated as second-order cone programming (SOCP) relaxation with maximal clique- and cycle-based SDP feasibility constraints. The formulation is based on the positive semi-definite (PSD) matrix completion theorem, which states that if all sub-matrices corresponding to maximal cliques in a chordal graph are PSD, then the partial matrix related to the chordal graph can be completed as a full PSD matrix. Existing methods in [1] first construct a chordal graph through Cholesky factorization. In this paper, we identify maximal cliques and minimal chordless cycles first. Enforcing the submatrices related to the maximal cliques and cycles PSD will guarantee a PSD full matrix. Further, we conduct chordal relaxation for the minimal chordless cycles by adding virtual lines and decompose each chordless cycle to 3-node cycles. Thus, the entire g
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We consider the problem of estimating an unknown matrix $\bX\in \reals^{m\times n}$, from observations $\bY = \bX+\bW$ where $\bW$ is a noise matrix with independent and identically distributed entries, as to minimize estimation error measured in operator norm. Assuming that the underlying signal $\bX$ is low-rank and incoherent with respect to the canonical basis, we prove that minimax risk is equivalent to $(\sqrt{m}\vee\sqrt{n})/\sqrt{\Info_W}$ in the high-dimensional limit $m,n\to\infty$, where $\Info_W$ is the Fisher information of the noise. Crucially, we develop an efficient procedure that achieves this risk, adaptively over the noise distribution (under certain regularity assumptions). Letting $\bX = \bU\bSigma\bV^{\sT}$ --where $\bU\in \reals^{m\times r}$, $\bV\in\reals^{n\times r}$ are orthogonal, and $r$ is kept fixed as $m,n\to\infty$-- we use our method to estimate $\bU$, $\bV$. Standard spectral methods provide non-trivial estimates of the factors $\bU,\bV$ (weak recovery
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Testing two potentially multivariate variables for statistical dependence on the basis finite samples is a fundamental statistical challenge. Here we explore a family of tests that adapt to the complexity of the relationship between the variables, promising robust power across scenarios. Building on the distance correlation, we introduce a family of adaptive independence criteria based on nonlinear monotonic transformations of distances. We show that these criteria, like the distance correlation and RKHS-based criteria, provide dependence indicators. We propose a class of adaptive (multi-threshold) test statistics, which form the basis for permutation tests. These tests empirically outperform some of the established tests in average and worst-case statistical sensitivity across a range of univariate and multivariate relationships and might deserve further exploration.
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An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f:E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number of $G$, denoted by $\chi_{la}(G)$, is the minimum number of distinct induced vertex labels over all local antimagic labelings of $G$. In this paper, we give counterexamples to the lower bound of $\chi_{la}(G \vee O_2)$ that was obtained in [Local antimagic vertex coloring of a graph, Graphs and Combin., 33 : 275 - 285 (2017)]. A sharp lower bound of $\chi_{la}(G\vee O_n)$ and sufficient conditions for the given lower bound to be attained are obtained. Moreover, we settled Theorem 2.15 and solved Problem 3.3 in the affirmative.
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We give an introduction to the study of algebraic hypersurfaces, focusing on the problem of when two hypersurfaces are isomorphic or close to being isomorphic. Working with hypersurfaces and emphasizing examples makes it possible to discuss these questions without any previous knowledge of algebraic geometry. At the end we formulate the main recent results and state the most important open questions.
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Many problems arise in computational biology can be reduced to the minimization of energy function, that determines on the geometry of considered molecule. The solution of this problem allows in particular to solve folding and docking problems in structural biology. For the small molecules this problem is well solved. But for the large molecules ($10^4$ atoms and more) this is still an open problem. In this work we consider energy minimization problem (OPLS force field) for the large molecules but with good enough initial (starting) point. In the paper one can find a biological explanation of this assumption. Due to this assumption we reduce the global optimization problem to the local one. We compare different methods: gradient-free methods, gradient type methods (gradient method, fast gradient method, conjugate gradients (CG), LBFGS), high-order (tensor) methods. We observe that the most convenient ones in GPU realization are fast gradient descent with special line-search and CG (Pol
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Data centres that use consumer-grade disks drives and distributed peer-to-peer systems are unreliable environments to archive data without enough redundancy. Most redundancy schemes are not completely effective for providing high availability, durability and integrity in the long-term. We propose alpha entanglement codes, a mechanism that creates a virtual layer of highly interconnected storage devices to propagate redundant information across a large scale storage system. Our motivation is to design flexible and practical erasure codes with high fault-tolerance to improve data durability and availability even in catastrophic scenarios. By flexible and practical, we mean code settings that can be adapted to future requirements and practical implementations with reasonable trade-offs between security, resource usage and performance. The codes have three parameters. Alpha increases storage overhead linearly but increases the possible paths to recover data exponentially. Two other paramet
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A finite word $w$ of length $n$ contains at most $n+1$ distinct palindromic factors. If the bound $n+1$ is reached, the word $w$ is called rich. An infinite word $w$ is called rich if every finite factor of $w$ is rich. Let $w$ be a rich word (finite or infinite) over an alphabet with $q>1$ letters, let $F(w,n)$ be the set of factors of length $n$ of the word $w$ and let $F_p(w,n)\subseteq F(w,n)$ be the set of palindromic factors of length $n$ of the word $w$. We show that $\vert F_p(w,n)\vert\leq (q+1)n(4q^{10}n)^{\log_2{n}}$ and $\vert F(w,n)\vert \leq (q+1)^2n^4(4q^{10}n)^{2\log_2{n}}$. It is known that $\vert F_p(w,n)\vert +\vert F_p(w,n+1)\vert \leq \vert F(w,n+1)\vert-\vert F(w,n)\vert+2$, where $w$ is an infinite word closed under reversal [Bal\'a\v{z}i, Mas\'akov\'a, Pelantov\'a, Theor. Comput. Sci., 380 (2007)]. We generalize this inequality for finite words and consequently we derive that $\vert F(w,n)\vert \leq 2(n-1)\hat F_p(w,n)-2(n-1) +q$ and $\vert F(w,n)\vert \leq 2
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We enrich the classical count that there are two complex lines meeting four lines in space to an equality of isomorphism classes of bilinear forms. For any field $k$, this enrichment counts the number of lines meeting four lines defined over $k$ in $\mathbb{P}^3_k$, with such lines weighted by their fields of definition together with information about the cross-ratio of the intersection points and spanning planes. We generalize this example to an infinite family of such enrichments, obtained using an Euler number in $\mathbb{A}^1$-homotopy theory. The classical counts are recovered by taking the rank of the bilinear forms.
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In Benzi & Olshanskii (SIAM J.~Sci.~Comput., 28(6) (2006)) a preconditioner of augmented Lagrangian type was presented for the two-dimensional stationary incompressible Navier--Stokes equations that exhibits convergence almost independent of Reynolds number. The algorithm relies on a highly specialized multigrid method involving a custom prolongation operator and is tightly coupled to the use of piecewise constant finite elements for the pressure. However, the prolongation operator and velocity element used do not directly extend to three dimensions: the local solves necessary in the prolongation operator do not satisfy the inf-sup condition. In this work we generalize the preconditioner to three dimensions, proposing alternative finite elements for the velocity and prolongation operators for which the preconditioner works robustly. The solver is effective at high Reynolds number: on a three-dimensional lid-driven cavity problem with approximately one billion degrees of freedom, th
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As an extension of our previous work in Sun et.al (2018) [41], we develop a discontinuous Galerkin method for solving cross-diffusion systems with a formal gradient flow structure. These systems are associated with non-increasing entropy functionals. For a class of problems, the positivity (non-negativity) of solutions is also expected, which is implied by the physical model and is crucial to the entropy structure. The semi-discrete numerical scheme we propose is entropy stable. Furthermore, the scheme is also compatible with the positivity-preserving procedure in Zhang (2017) [42] in many scenarios. Hence the resulting fully discrete scheme is able to produce non-negative solutions. The method can be applied to both one-dimensional problems and two-dimensional problems on Cartesian meshes. Numerical examples are given to examine the performance of the method.
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In this work the authors consider an inverse source problem in the following stochastic fractional diffusion equation $$\partial_t^\alpha u(x,t)+\mathcal{A} u(x,t)=f(x)h(t)+g(x) \dot{\mathbb{W}}(t).$$ The interested inverse problem is to reconstruct $f(x)$ and $g(x)$ by the statistics of the final time data $u(x,T).$ Some direct problem results are proved at first, such as the existence, uniqueness, representation and regularity of the solution. Then the reconstruction scheme for $f$ and $g$ is given. To tackle the ill-posedness, the Tikhonov regularization is adopted. Finally we give a regularized reconstruction algorithm and some numerical results are displayed.
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We provide novel sufficient conditions for the uniform recovery of sparse Legendre expansions using $\ell_1$ minimization, where the sampling points are drawn according to orthogonalization (uniform) measure. So far, conditions of the form $m \gtrsim \Theta^2 s \times \textit{log factors}$ have been relied on to determine the minimum number of samples $m$ that guarantees successful reconstruction of $s$-sparse vectors when the measurement matrix is associated to an orthonormal system. However, in case of sparse Legendre expansions, the uniform bound $\Theta$ of Legendre systems is so high that these conditions are unable to provide meaningful guarantees. In this paper, we present an analysis which employs the envelop bound of all Legendre polynomials instead, and prove a new recovery guarantee for $s$-sparse Legendre expansions, $$ m \gtrsim {s^2} \times \textit{log factors}, $$ which is independent of $\Theta$. Arguably, this is the first recovery condition established for orthonormal
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Gradient descent (GD) converges linearly to the global optimum for even nonconvex problems when the loss function satisfies certain benign geometric properties that are strictly weaker than strong convexity. One important property studied in the literature is the so-called Regularity Condition (RC). The RC property has been proven valid for many problems such as deep linear neural networks, shallow neural networks with nonlinear activations, phase retrieval, to name a few. Moreover, accelerated first-order methods (e.g. Nesterov's accelerated gradient and Heavy-ball) achieve great empirical success when the parameters are tuned properly but lack theoretical understandings in the nonconvex setting. In this paper, we use tools from robust control to analytically characterize the region of hyperparameters that ensure linear convergence of accelerated first-order methods under RC. Our results apply to all functions satisfying RC and therefore are more general than results derived for speci
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In this paper, we focus on approaches to parallelizing stochastic gradient descent (SGD) wherein data is farmed out to a set of workers, the results of which, after a number of updates, are then combined at a central master node. Although such synchronized SGD approaches parallelize well in idealized computing environments, they often fail to realize their promised computational acceleration in practical settings. One cause is slow workers, termed stragglers, who can cause the fusion step at the master node to stall, which greatly slowing convergence. In many straggler mitigation approaches work completed by these nodes, while only partial, is discarded completely. In this paper, we propose an approach to parallelizing synchronous SGD that exploits the work completed by all workers. The central idea is to fix the computation time of each worker and then to combine distinct contributions of all workers. We provide a convergence analysis and optimize the combination function. Our numeric
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Gaussian random vectors exhibit the loss of dimension phenomena, which relate to their joint survival tail behaviour. Besides, the fact that the components of such vectors are light-tailed complicates the approximations of various multivariate risk measures significantly. In this contribution we derive precise approximations of marginal mean excess, marginal expected shortfall and multivariate conditional tail expectation of Gaussian random vectors and highlight links with conditional limit theorems. Our study indicates that similar results hold for elliptical and Gaussian like multivariate risks.
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The Askey--Wilson polynomials are the most general classical orthogonal polynomials that are known and the Nassrallah--Rahman integral is a very general extension of Euler's integral representation of the classical $_2F_1$ function. Based on a $q$-series transformation formula and the Nassrallah--Rahman integral we prove a $q$--beta integral which has twelve parameters, with several other results, both classical and new, included as special cases. This $q$-beta integral also allows us to derive a curious double $q$--series transformation formula, which includes one formula of Al--Salam and Ismail as a special case
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We show that the asymptotic dimension of a hyperbolic relatively hyperbolic graph is finite provided that this holds true uniformly for the peripheral subgraphs and for the electrifiation. We use this to show that the asymptotic dimension of the disk graph of a handlebody of genus at least two is at most quadratic in the genus.
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We study the problem of optimal observability and prove time asymptotic observability estimates for the Schr\"odinger equation with a potential in $L^{\infty}(\Omega)$, with $\Omega\subset \mathbb{R}^d$, using spectral theory. An elegant way to model the problem using a time asymptotic observability constant is presented. For certain small potentials, we demonstrate the existence of a nonzero asymptotic observability constant under given conditions and describe its explicit properties and optimal values. Moreover, we give a precise description of numerical models to analyze the properties of important examples of potentials wells, including that of the modified harmonic oscillator.
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Let $(R,M)$ be a quasilocal integral domain. We investigate the set of irreducible elements (atoms) of $R$. Special attention is given to the set of atoms in $M \backslash M^2$ and to the existence of atoms in $M^2$. While our main interest is in local Cohen-Kaplansky (CK) domains (atomic integral domains with only finitely many non-associate atoms), we endeavor to obtain results in the greatest generality possible. In contradiction to a statement of Cohen and Kaplansky, we construct a local CK domain with precisely eight nonassociate atoms having an atom in $M^2$.
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If $X$ is the attractor set of a conformal IFS in dimension two or three, we prove that there exists a quasiregular semigroup $G$ with Julia set equal to $X$. We also show that in dimension two, with a further assumption similar to the open set condition, the same result can be achieved with a semigroup generated by one element. Consequently, in this case the attractor set is quasiconformally equivalent to the Julia set of a rational map.
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We consider BSDEs with two reflecting irregular barriers. We give necessary and sufficient conditions for existence and uniqueness of $\mathbb{L}^{p}$ solutions for equations with generators monotone with respect to $y$ and Lipschitz continuous with respect to $z$, and with data in $\mathbb{L}^{p}$ spaces for $p\ge 1$. We also prove that the solutions can be approximated via penalization method.
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In this paper we use the theory of barcodes as a new tool for studying dynamics of area-preserving homeomorphisms. We will show that the barcode of a Hamiltonian diffeomorphism of a surface depends continuously on the diffeomorphism, and furthermore define barcodes for Hamiltonian homeomorphisms. Our main dynamical application concerns the notion of {\it weak conjugacy}, an equivalence relation which arises naturally in connection to $C^0$ continuous conjugacy invariants of Hamiltonian homeomorphisms. We show that for a large class of Hamiltonian homeomorphisms with a finite number of fixed points, the number of fixed points, counted with multiplicity, is a weak conjugacy invariant. The proof relies, in addition to the theory of barcodes, on techniques from surface dynamics such as Le Calvez's theory of transverse foliations. In our exposition of barcodes and persistence modules, we present a proof of the Isometry Theorem which incorporates Barannikov's theory of simple Morse complexes
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A notion of a split quasi-hereditary algebra has been defined by Cline, Parshall and Scott. Du and Rui describe a based approach to split quasi-hereditary algebras. We develop this approach further to show that over a complete local Noetherian ring, one can achieve even stronger basis properties. This is important for `schurifying' quasi-hereditary algebras as developed in our subsequent work. The schurification procedure associates to an algebra $A$ a new algebra, which is the classical Schur algebra if $A$ is a field. Schurification produces interesting new quasi-hereditary and cellular algebras. It is important to work over an integral domain of characteristic zero, taking into account a super-structure on the input algebra $A$. So we pay attention to super-structures on quasi-hereditary algebras and investigate a subtle conforming property of heredity data which is crucial to guarantee that the schurification of $A$ is quasi-hereditary if so is $A$. We establish a Morita equivalenc
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We construct the anticyclotomic $p$-adic $L$-function that interpolates a square root of central values of twisted spinor $L$-functions of a quadratic base change of a Siegel cusp form of genus $2$ with respect to a paramodular group of square-free level, assuming the B\"ocherer conjecture for the central $L$-values with anticyclotomic twists.
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In \cite{GS1} the notion of braided Yangians of Reflection Equation type was introduced. Each of these algebras is associated with an involutive or Hecke symmetry $R$. Besides, the quantum analogs of certain symmetric polynomials (elementary symmetric ones, power sums) were suggested. In the present paper we show that these quantum symmetric polynomials commute with each other and consequently generate a commutative Bethe subalgebra. As an application, we get some Gaudin-type models and the corresponding Bethe subalgebras.
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Hurwitz numbers enumerate ramified coverings of the Riemann sphere with fixed ramification data. Certain kinds of ramification data are of particular interest, such as double Hurwitz numbers, which count covers with fixed arbitrary ramification over $0$ and $\infty$ and simple ramification over $b$ points, where $b$ is given by the Riemann-Hurwitz formula. In this work, we introduce the notion of bi-pruned double Hurwitz numbers. This is a new enumerative problem, which yields smaller numbers but completely determines double Hurwitz numbers. They count a relevant subset of covers and share many properties with double Hurwitz numbers, such as piecewise polynomial behaviour and an expression in the symmetric group. Thus, we may view them as a core of the double Hurwitz numbers problem. This work is built on and generalises previous work of Do--Norbury and the author.
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In this paper, we generalize the Picard-Fuchs equation method to study the bifurcation of limit cycles of perturbed differential systems with two switching lines. We obtain the detailed expression of the corresponding first order Melnikov function which can be used to get the upper bound of the number of limit cycles for the perturbed system by using Picard-Fuchs equation. It is worth noting that we greatly simplify the computations and this method can be applied to study the number of limit cycles of other differential systems with two switching lines. Our results also show that the number of switching lines has essentially impact on the number of limit cycles bifurcating from a period annulus.
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By using the Picard-Fuchs equation and the property of Chebyshev space to the discontinuous differential system, we obtain an upper bound of the number of limit cycles for the nongeneric quadratic reversible system when it is perturbed inside all discontinuous polynomials with degree $n$.
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For any surface $\Sigma$ of infinite topological type, we study the Torelli subgroup $\mathcal{I}(\Sigma)$ of the mapping class group ${\rm Mod}(\Sigma)$, whose elements are those mapping classes that act trivially on the homology of $\Sigma$. Our first result asserts that $\mathcal{I}(\Sigma)$ is topologically generated by the subgroup of ${\rm Mod}(\Sigma)$ consisting of those elements which have compact support. In particular, using results of Birman and Powell, we deduce that $\mathcal{I}(\Sigma)$ is topologically generated by {\em separating twists} and {\em bounding pair maps}. Next, we prove that the abstract commensurator of $\mathcal{I}(\Sigma)$ coincides with ${\rm Mod}(\Sigma)$. This extends results of Farb-Ivanov and Kida to the setting of infinite--type surfaces.
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We show that the Helmholtz equation describing the propagation of transverse electric waves in a Kerr slab with a complex linear permittivity and a complex Kerr coefficient admits blow-up solutions provided that the real part of the Kerr coefficient is negative, i.e., the slab is defocusing. This result applies to homogeneous as well as inhomogeneous Kerr slabs whose linear permittivity and Kerr coefficient are continuous functions of the transverse coordinate. For an inhomogeneous Kerr slab, blow-up solutions exist if the real part of Kerr coefficient is bounded above by a negative number.
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We study a statistical method to estimate the optimal value, and the optimality gap of a given solution for stochastic optimization as an assessment of the solution quality. Our approach is based on bootstrap aggregating, or bagging, resampled sample average approximation (SAA). We show how this approach leads to valid statistical confidence bounds for non-smooth optimization. We also demonstrate its statistical efficiency and stability that are especially desirable in limited-data situations, and compare these properties with some existing methods. We present our theory that views SAA as a kernel in an infinite-order symmetric statistic, which can be approximated via bagging. We substantiate our theoretical findings with numerical results.
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We study the persistence exponent for the first passage time of a random walk below the trajectory of another random walk. More precisely, let $\{B_n\}$ and $\{W_n\}$ be two centered, weakly dependent random walks. We establish that $\mathbb{P}(\forall_{n\leq N} B_n \geq W_n|W) = N^{-\gamma + o(1)}$ for a non-random $\gamma\geq 1/2$. In the classical setting, $W_n \equiv 0$, it is well-known that $\gamma = 1/2$. We prove that for any non-trivial $W$ one has $\gamma>1/2$ and the exponent $\gamma$ depends only on $\text{Var}(B_1)/\text{Var}(W_1)$. Our result holds also in the continuous setting, when $B$ and $W$ are independent and possibly perturbed Brownian motions or Ornstein-Uhlenbeck processes. In the latter case the probability decays at exponential rate.
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We consider random walk in a space-time random potential, also known as directed random polymer measures, on the planer square lattice with nearest-neighbor steps and general i.i.d. weights on the vertices. We construct covariant cocycles and use them to prove new results on existence, uniqueness/non-uniqueness, and asymptotic directions of semi-infinite polymer measures (solutions to the Dobrushin-Lanford-Ruelle equations). We also prove non-existence of covariant or deterministically directed bi-infinite polymer measures. Along the way, we prove almost sure existence of Busemann function limits in directions where the limiting free energy has some regularity.
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We consider the Cauchy problem for a model of non-linear acoustics, named the Kuznetsov equation, describing sound propagation in thermo-viscous elastic media. For the viscous case, it is a weakly quasi-linear strongly damped wave equation, for which we prove the global existence in time of regular solutions for sufficiently small initial data, the size of which is specified, and give the corresponding energy estimates. In the non-viscous case, we update the known results of John for quasi-linear wave equations, obtaining the well-posedness results for less regular initial data. We obtain, using a priori estimates and a Klainerman inequality, the estimations of the maximal existence time, depending on the space dimension, which are optimal, thanks to the blow-up results of Alinhac. Alinhac's blow-up results are also confirmed by a L 2-stability estimate, obtained between a regular and a less regular solutions.
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Answering a question of Sakai, we show that the existence of an $\omega_1$-Erd\H{o}s cardinal suffices to obtain the consistency of Chang's Conjecture with $\square_{\omega_1, 2}$. By a result of Donder this is best possible. We also give an answer to another question of Sakai relating to the incompatibility of $\square_{\lambda, 2}$ and $(\lambda^+, \lambda) \twoheadrightarrow (\kappa^+, \kappa)$ for uncountable $\kappa$.
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We show that a necessary and sufficient condition for a smooth function on the tangent bundle of a manifold to be a Lagrangian density whose action can be minimized is, roughly speaking, that it be the sum of a constant, a nonnegative function vanishing on the support of the minimizers, and an exact form. We show that this exact form corresponds to the differential of a Lipschitz function on the manifold that is differentiable on the projection of the support of the minimizers, and its derivative there is Lipschitz. This function generalizes the notion of subsolution of the Hamilton-Jacobi equation that appears in weak KAM theory, and the Lipschitzity result allows for the recovery of Mather's celebrated 1991 result as a special case. We also show that our result is sharp with several examples. Finally, we apply the same type of reasoning to an example of a finite horizon Legendre problem in optimal control, and together with the Lipschitzity result we obtain the Hamilton-Jacobi-Bellma
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Topological indices are real numbers invariant under graph isomorphisms. Chromatic analogue of topological indices has been introduced recently in literature in 2017. Mainly, chromatic versions of Zagreb indices are studied lately. This paper discusses the notion of chromatic topological and irregularity indices of certain cycle related graphs.
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Computably enumerable equivalence relations (ceers) received a lot of attention in the literature. The standard tool to classify ceers is provided by the computable reducibility $\leq_c$. This gives rise to a rich degree-structure. In this paper, we lift the study of $c$-degrees to the $\Delta^0_2$ case. In doing so, we rely on the Ershov hierarchy. For any notation $a$ for a non-zero computable ordinal, we prove several algebraic properties of the degree-structure induced by $\leq_c$ on the $\Sigma^{-1}_{a}\smallsetminus \Pi^{-1}_a$ equivalence relations. A special focus of our work is on the (non)existence of infima and suprema of $c$-degrees.
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When there is no independence, abnormal observations may have a tendency to appear in clusters instead of scattered along the time frame. Identifying clusters and estimating their size are important problems arising in statistics of extremes or in the study of quantitative recurrence for dynamical systems. In the classical literature, the Extremal Index appears associated to the cluster size and, in fact, it is usually interpreted as the reciprocal of the mean cluster size. This quantity involves a passage to the limit and in some special cases this interpretation fails due to an escape of mass when computing the limiting point processes. Smith \cite{S88} introduced a regenerative process exhibiting such disagreement. Very recently, in \cite{AFF18} the authors used a dynamical mechanism to emulate the same inadequacy of the usual interpretation of the Extremal Index. Here, we consider a general regenerative process that includes Smith's model and show that it is important to consider f
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We give strengthened versions of the Herwig-Lascar and Hodkinson-Otto extension theorems for partial automorphisms of finite structures. Such strengthenings yield several combinatorial and group-theoretic consequences for homogeneous structures. For instance, we establish a coherent form of the extension property for partial automorphisms for certain Fraisse classes. We deduce from these results that the isometry group of the rational Urysohn space, the automorphism group of the Fraisse limit of any Fraisse class that is the class of all $\mathcal{F}$-free structures (in the Herwig--Lascar sense), and the automorphism group of any free homogeneous structure over a finite relational language, all contain a dense locally finite subgroup. We also show that any free homogeneous structure admits ample generics.
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In a recent work, Andrews gave analytic proofs of two conjectures concerning some variations of two combinatorial identities between partitions of a positive integer into odd parts and partitions into distinct parts discovered by Beck. Subsequently, using the same method as Andrews, Chern presented the analytic proof of another Beck's conjecture relating the gap-free partitions and distinct partitions with odd length. However, the combinatorial interpretations of these conjectures are still unclear and required. In this paper, motivated by Glaisher's bijection, we give the combinatorial proofs of these three conjectures directly or by proving more generalized results.
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We compare the degrees of enumerability and the closed Medvedev degrees and find that many situations occur. There are nonzero closed degrees that do not bound nonzero degrees of enumerability, there are nonzero degrees of enumerability that do not bound nonzero closed degrees, and there are degrees that are nontrivially both degrees of enumerability and closed degrees. We also show that the compact degrees of enumerability exactly correspond to the cototal enumeration degrees.
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In this paper, we provide a negative answer to a long-standing open problem on the compatibility of Spearman's rho matrices. Following an equivalence of Spearman's rho matrices and linear correlation matrices for dimensions up to 9 in the literature, we show non-equivalence for dimensions 12 or higher. In particular, we connect this problem with the existence of a random vector under some linear projection restrictions in two characterization results.
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In this paper we consider a system of parabolic reaction-diffusion equations with strong competition and two related scalar reaction-diffusion equations. We are mainly concerned with the case of periodic coefficients and periodic solutions. We show that, for sufficiently large periods, these models have stationary, non-constant, fully non-trivial and stable solutions. We compare our results with already known results about the existence and non-existence of such solutions. Finally, we provide ecological interpretations for these results in terms of resistance against an invasion.
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Automorphism groups are intrincate conjugacy invariants for subshifts, which can reveal important features of the dynamical structure of a shift action. One important case is the study of automorphism groups when the underlying subshift has a very rigid structure, e.g. substitutive subshifts or aperiodic constructions with large-scale self-similarity, such as the Robinson shift. In this work we study the automorphism group of bijective substitutive subshifts, and a potential generalization in the form of the group of extended symmetries, studied previously by Michael Baake, John Roberts and Reem Yassawi (arXiv:1611.05756); these symmetries, by allowing for shearing and other deformations of the underlying group, may reveal additional information of a geometric nature about the structure of these subshifts.
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We study the convex hulls of trajectories of polynomial dynamical systems. Such trajectories include real algebraic curves. The boundaries of the resulting convex bodies are stratified into families of faces. We present numerical algorithms for identifying these patches. An implementation based on the software Bensolve Tools is given. This furnishes a key step in computing attainable regions of chemical reaction networks.
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We show that for a conformal local net of observables on the circle, the split property is automatic. Both full conformal covariance (i.e. diffeomorphism covariance) and the circle-setting play essential roles in this fact, while by previously constructed examples it was already known that even on the circle, M\"obius covariance does not imply the split property. On the other hand, here we also provide an example of a local conformal net living on the two-dimensional Minkowski space, which - although being diffeomorphism covariant - does not have the split property.
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In a group $G$, elements $a$ and $b$ are conjugate if there exists $g\in G$ such that $g^{-1} ag=b$. This conjugacy relation, which plays an important role in group theory, can be extended in a natural way to inverse semigroups: for elements $a$ and $b$ in an inverse semigroup $S$, $a$ is conjugate to $b$, which we will write as $a\sim_{\mathrm{i}} b$, if there exists $g\in S^1$ such that $g^{-1} ag=b$ and $gbg^{-1} =a$. The purpose of this paper is to study the conjugacy $\sim_{\mathrm{i}}$ in several classes of inverse semigroups: symmetric inverse semigroups, free inverse semigroups, McAllister $P$-semigroups, factorizable inverse monoids, Clifford semigroups, the bicyclic monoid and stable inverse semigroups.
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This paper proves the consistency property for the regularized maximum likelihood estimators (MLEs) of multivariate Hawkes processes (MHPs). It also develops an alternating minimization type algorithm (AA-iPALM) to compute the MLEs with guaranteed global convergence to the set of stationary points. The performance of this AA-iPALM algorithm on both synthetic and real-world data shows that AA-iPALM consistently improves over iPALM and PALM. Moreover, AA-iPALM is able to identify the causality structure in rental listings on craigslist and herd behavior in the NASDAQ ITCH dataset.
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We define biquandle structures on a given quandle, and show that any biquandle is given by some biquandle structure on its underlying quandle. We characterize all biquandles with a given underlying quandle. Using this characterization, we obtain a relationship between the automorphism group of a biquandle and the automorphism group of its underlying quandle. As an application, we determine the automorphism groups of Alexander and dihedral biquandles. We also define a biquandle structure on a direct product of two quandles and describe the automorphism group of the biquandle thus obtained.
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We correct a mistake regarding almost complex structures on Hilbert schemes of points in surfaces in arXiv:1510.02449. The error does not affect the main results of the paper, and only affects the proofs of invariance of equivariant symplectic Khovanov homology and reduced symplectic Khovanov homology. We give an alternate proof of the invariance of equivariant symplectic Khovanov homology.
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In this paper, we derive an asymptotic formula for the number of conjugacy classes of elements in a class of statistically convex-cocompact actions with contracting elements. Denote by $\mathcal C(n)$ (resp. $\mathcal C'(n)$) the set of (resp. primitive) conjugacy classes of stable length at most $n$. The main result is an asymptotic formula as follows: $$\sharp \mathcal C(n) \asymp \sharp \mathcal C'(n) \asymp \frac{\exp(\e G)}{n}.$$ The same formula holds for conjugacy classes using pointed length for any given basepoint. As a consequence of the formulae, the conjugacy growth series is transcendental for all non-elementary relatively hyperbolic groups, graphical small cancellation groups with finite components. As by-product of the proof, we establish several useful properties for an exponentially generic set of elements. In particular, it yields a positive answer to a question of Maher that an exponentially generic elements in mapping class groups have their Teichm\"{u}ller axis con
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Counting functions are constructed for sums of integers raised to a fixed positive rational power. That is, given values formed by $u_1^{j/k} + u_2^{j/k} + ... + u_l^{j/k}$, $u_i \in \mathbb{Z}^+$, the number of values less than or equal to a given $w>0$ is determined. The counting functions developed are framed in terms of convolution exponentials, and are closely related to the Riemann zeta function. At the conclusion, several estimates are derived, with special emphasis on the case of sums of square roots, i.e. $j=1$, $k=2$.
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This paper considers a single cell multi-antenna base station delivering content to multiple cache enabled single-antenna users. Coding strategies are developed that allow for decentralized placement in the wireless setting. Three different cases namely, max-min multicasting, linear combinations in the complex field, and linear combinations in the finite field, are considered and closed-form rate expressions are provided that hold with high probability. For the case of max-min fair multicasting delivery, we propose a new coding scheme that is capable of working with only two-user broadcasts. A cyclic-exchange protocol for efficient content delivery is proposed and shown to perform almost as well as the original multi-user broadcast scheme.
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One of the classical results concerning differentiability of continuous functions states that the set $\mathcal{SD}$ of somewhere differentiable functions (i.e., functions which are differentiable at some point) is Haar-null in the space $C[0,1]$. By a recent result of Banakh et al., a set is Haar-null provided that there is a Borel hull $B\supseteq A$ and a continuous map $f\colon \{0,1\}^\mathbb{N}\to C[0,1]$ such that $f^{-1}[B+h]$ is Lebesgue's null for all $h\in C[0,1]$. We prove that $\mathcal{SD}$ is not Haar-countable (i.e., does not satisfy the above property with "Lebesgue's null" replaced by "countable", or, equivalently, for each copy $C$ of $\{0,1\}^\mathbb{N}$ there is an $h\in C[0,1]$ such that $\mathcal{SD}\cap (C+h)$ is uncountable. Moreover, we use the above notions in further studies of differentiability of continuous functions. Namely, we consider functions differentiable on a set of positive Lebesgue's measure and functions differentiable almost everywhere with res
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This paper studies the consensusability problem of multi-agent systems (MASs), where agents communicate with each other through Markovian packet loss channels. We try to determine conditions under which there exists a linear distributed consensus controller such that the MAS can achieve mean square consensus. We first provide a necessary and sufficient consensus condition for MASs with single input and i.i.d.\ channel losses, which complements existing results. Then we proceed to study the case with identical Markovian packet losses. A necessary and sufficient consensus condition is firstly derived based on the stability of Markov jump linear systems. Then a numerically verifiable consensus criterion in terms of the feasibility of linear matrix inequalities (LMIs) is proposed. Furthermore, analytic sufficient conditions and necessary conditions for mean square consensusability are provided for general MASs. The case with nonidentical packet loss is studied subsequently. The necessary a
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A detection system with a single sensor and two detectors is considered, where each of the terminals observes a memoryless source sequence, the sensor sends a message to both detectors and the first detector sends a message to the second detector. Communication of these messages is assumed to be error-free but rate-limited. The joint probability mass function (pmf) of the source sequences observed at the three terminals depends on an $\mathsf{M}$-ary hypothesis $(\mathsf{M} \geq 2)$, and the goal of the communication is that each detector can guess the underlying hypothesis. Detector $k$, $k=1,2$, aims to maximize the error exponent \textit{under hypothesis} $i_k$, $i_k \in \{1,\ldots,\mathsf{M}\}$, while ensuring a small probability of error under all other hypotheses. We study this problem in the case in which the detectors aim to maximize their error exponents under the \textit{same} hypothesis (i.e., $i_1=i_2$) and in the case in which they aim to maximize their error exponents und
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