In this paper, we discuss Cameron-Liebler sets in Hamming graphs, obtain several equivalent definitions and present all classification results.

For a smooth projective variety $X$, we study analogs of Quot functors in hearts of non-standard $t$-structures of $D^b(mathrm{Coh}(X))$. The technical framework is that of families of $t$-structures, as studied in arXiv:1902.08184. We provide several examples and suggest possible directions of further investigation, as we reinterpret moduli spaces of stable pairs, in the sense of Thaddeus (arXiv:alg-geom/9210007) and Huybrechts-Lehn (arXiv:alg-geom/9211001), as instances of Quot schemes.

We consider a nonlinear Dirichlet problem driven by a nonhomogeneous differential operator plus an indefinite potential. In the reaction we have the competing effects of a singular term and of concave and convex nonlinearities. In this paper the concave term is parametric. We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the positive parameter $lambda$ varies. This work continues our research published in arXiv:2004.12583, where $xi equiv 0 $ and in the reaction the parametric term is the singular one.

We show that the complex absorbing potential (CAP) method for computing scattering resonances applies to the case of exponentially decaying potentials. That means that the eigenvalues of $-Delta + V - iepsilon x^2$, $|V(x)|leq e^{-2gamma |x|}$ converge, as $ epsilon o 0+ $, to the poles of the meromorphic continuation of $ ( -Delta + V -lambda^2 )^{-1} $ uniformly on compact subsets of $ extrm{Re},lambda>0$, $ extrm{Im},lambda>-gamma$, $arglambda > pi/8$.

In this paper, for a family of second-order parabolic equation with rapidly oscillating and time-dependent periodic coefficients, we are interested in an approximate two-sphere one-cylinder inequality for these solutions in parabolic periodic homogenization, which implies an approximate quantitative propagation of smallness. The proof relies on the asymptotic behavior of fundamental solutions and the Lagrange interpolation technique.

An inverse spectral problem for the Sturm-Liouville operator with a singular potential from the class $W_2^{-1}$ is solved by the method of spectral mappings. We prove the uniqueness theorem, develop a constructive algorithm for solution, and obtain necessary and sufficient conditions of solvability for the inverse problem in the self-adjoint and the non-self-adjoint cases

We construct a one-to-one continuous map from the Morse boundary of a hierarchically hyperbolic group to its Martin boundary. This construction is based on deviation inequalities generalizing Ancona's work on hyperbolic groups. This provides a possibly new metrizable topology on the Morse boundary of such groups. We also prove that the Morse boundary has measure 0 with respect to the harmonic measure unless the group is hyperbolic.

In this paper, we mainly establish the existence and uniqueness theorem for solutions of the exterior Dirichlet problem for a class of fully nonlinear second-order elliptic equations related to the eigenvalues of the Hessian, with prescribed generalized symmetric asymptotic behavior at infinity. Moreover, we give some new results for the Hessian equations, Hessian quotient equations and the special Lagrangian equations, which have been studied previously.

We show how to efficiently compute the derivative (when it exists) of the solution map of log-log convex programs (LLCPs). These are nonconvex, nonsmooth optimization problems with positive variables that become convex when the variables, objective functions, and constraint functions are replaced with their logs. We focus specifically on LLCPs generated by disciplined geometric programming, a grammar consisting of a set of atomic functions with known log-log curvature and a composition rule for combining them. We represent a parametrized LLCP as the composition of a smooth transformation of parameters, a convex optimization problem, and an exponential transformation of the convex optimization problem's solution. The derivative of this composition can be computed efficiently, using recently developed methods for differentiating through convex optimization problems. We implement our method in CVXPY, a Python-embedded modeling language and rewriting system for convex optimization. In just a few lines of code, a user can specify a parametrized LLCP, solve it, and evaluate the derivative or its adjoint at a vector. This makes it possible to conduct sensitivity analyses of solutions, given perturbations to the parameters, and to compute the gradient of a function of the solution with respect to the parameters. We use the adjoint of the derivative to implement differentiable log-log convex optimization layers in PyTorch and TensorFlow. Finally, we present applications to designing queuing systems and fitting structured prediction models.

Given graphs $T$ and $H$, the generalized Tur'an number ex$(n,T,H)$ is the maximum number of copies of $T$ in an $n$-vertex graph with no copies of $H$. Alon and Shikhelman, using a result of ErdH os, determined the asymptotics of ex$(n,K_3,H)$ when the chromatic number of $H$ is greater than 3 and proved several results when $H$ is bipartite. We consider this problem when $H$ has chromatic number 3. Even this special case for the following relatively simple 3-chromatic graphs appears to be challenging.

The suspension $widehat H$ of a graph $H$ is the graph obtained from $H$ by adding a new vertex adjacent to all vertices of $H$. We give new upper and lower bounds on ex$(n,K_3,widehat{H})$ when $H$ is a path, even cycle, or complete bipartite graph. One of the main tools we use is the triangle removal lemma, but it is unclear if much stronger statements can be proved without using the removal lemma.

Applying the equivariant moving frames method, we construct convergent normal forms for real-analytic 5-dimensional totally nondegenerate CR submanifolds of C^4. These CR manifolds are divided into several biholomorphically inequivalent subclasses, each of which has its own complete normal form. Moreover it is shown that, biholomorphically, Beloshapka's cubic model is the unique member of this class with the maximum possible dimension seven of the corresponding algebra of infinitesimal CR automorphisms. Our results are also useful in the study of biholomorphic equivalence problem between CR manifolds, in question.

The $(q, mathbf{Q})$-current algebra associated with the general linear Lie algebra was introduced by the second author in the study of representation theory of cyclotomic $q$-Schur algebras. In this paper, we study the $(q, mathbf{Q})$-current algebra $U_q(mathfrak{sl}_n^{langle mathbf{Q} angle}[x])$ associated with the special linear Lie algebra $mathfrak{sl}_n$. In particular, we classify finite dimensional simple $U_q(mathfrak{sl}_n^{langle mathbf{Q} angle}[x])$-modules.

In this article, we prove that the completeness of the Hamilton flow and essential self-dajointness are equivalent for real principal type operators on the circle. Moreover, we study spectral properties of these operators.

It is well-known that a celebrated J"{o}rgens-Calabi-Pogorelov theorem for Monge-Amp`ere equations states that any classical (viscosity) convex solution of $det(D^2u)=1$ in $mathbb{R}^n$ must be a quadratic polynomial. Therefore, it is an interesting topic to study the existence and uniqueness theorem of such fully nonlinear partial differential equations' Dirichlet problems on exterior domains with suitable asymptotic conditions at infinity. As a continuation of the works of Caffarelli-Li for Monge-Amp`ere equation and of Bao-Li-Li for $k$-Hessian equations, this paper is devoted to the solvability of the exterior Dirichlet problem of Hessian quotient equations $sigma_k(lambda(D^2u))/sigma_l(lambda(D^2u))=1$ for any $1leq l<kleq n$ in all dimensions $ngeq 2$. By introducing the concept of generalized symmetric subsolutions and then using the Perron's method, we establish the existence theorem for viscosity solutions, with prescribed asymptotic behavior which is close to some quadratic polynomial at infinity.

Consider that $u_0$ nodes are aware of some piece of data $d_0$. This note derives the expected time required for the data $d_0$ to be disseminated through-out a network of $n$ nodes, when communication between nodes evolves according to a graphical Markov model $overline{ mathcal{G}}_{n,hat{p}}$ with probability parameter $hat{p}$. In this model, an edge between two nodes exists at discrete time $k in mathbb{N}^+$ with probability $hat{p}$ if this edge existed at $k-1$, and with probability $(1-hat{p})$ if this edge did not exist at $k-1$. Each edge is interpreted as a bidirectional communication link over which data between neighbors is shared. The initial communication graph is assumed to be an Erdos-Renyi random graph with parameters $(n,hat{p})$, hence we consider a emph{stationary} Markov model $overline{mathcal{G}}_{n,hat{p}}$. The asymptotic "$u_0$-expected flooding time" of $overline{mathcal{G}}_{n,hat{p}}$ is defined as the expected number of iterations required to transmit the data $d_0$ from $u_0$ nodes to $n$ nodes, in the limit as $n$ approaches infinity. Although most previous results on the asymptotic flooding time in graphical Markov models are either emph{almost sure} or emph{with high probability}, the bounds obtained here are emph{in expectation}. However, our bounds are tighter and can be more complete than previous results.

We relate the $L^p$-mapping properties of the Bergman projections on two domains in $mathbb{C}^n$, one of which is the quotient of the other under the action of a finite group of biholomorphic automorphisms. We use this relation to deduce the sharp ranges of $L^p$-boundedness of the Bergman projection on certain $n$-dimensional model domains generalizing the Hartogs triangle.

In this paper we collect some open set-theoretic problems that appear in the large-scale topology (called also Asymptology). In particular we ask problems about critical cardinalities of some special (large, indiscrete, inseparated) coarse structures on $omega$, about the interplay between properties of a coarse space and its Higson corona, about some special ultrafilters ($T$-points and cellular $T$-points) related to finitary coarse structures on $omega$, about partitions of coarse spaces into thin pieces, and also about coarse groups having some extremal properties.

Here we prove that under a lower sectional curvature bound, a sequence of manifolds diffeomorphic to the standard $m$-dimensional torus cannot converge in the Gromov-Hausdorff sense to a closed interval.

We prove a continuity result for the shearlet transform when restricted to the space of smooth and rapidly decreasing functions with all vanishing moments. We define the dual shearlet transform, called here the shearlet synthesis operator, and we prove its continuity on the space of smooth and rapidly decreasing functions over $mathbb{R}^2 imesmathbb{R} imesmathbb{R}^ imes$. Then, we use these continuity results to extend the shearlet transform to the space of Lizorkin distributions, and we prove its consistency with the classical definition for test functions.

We show weak-strong uniqueness and stability results for the motion of a two or three dimensional fluid governed by the Navier-Stokes equation interacting with a flexible, elastic plate of Koiter type. The plate is situated at the top of the fluid and as such determines the variable part of a time changing domain (that is hence a part of the solution) containing the fluid. The uniqueness result is a consequence of a stability estimate where the difference of two solutions is estimated by the distance of the initial values and outer forces. For that we introduce a methodology that overcomes the problem that the two (variable in time) domains of the fluid velocities and pressures are not the same. The estimate holds under the assumption that one of the two weak solutions possesses some additional higher regularity. The additional regularity is exclusively requested for the velocity of one of the solutions resembling the celebrated Ladyzhenskaya-Prodi-Serrin conditions in the framework of variable domains.

Let $k ,=, mathbb{Q}(sqrt[5]{n},zeta_5)$, where $n$ is a positive integer, $5^{th}$ power-free, whose $5-$class group is isomorphic to $mathbb{Z}/5mathbb{Z} imesmathbb{Z}/5mathbb{Z}$. Let $k_0,=,mathbb{Q}(zeta_5)$ be the cyclotomic field containing a primitive $5^{th}$ root of unity $zeta_5$. Let $C_{k,5}^{(sigma)}$ the group of the ambiguous classes under the action of $Gal(k/k_0)$ = $<sigma>$. The aim of this paper is to determine all integers $n$ such that the group of ambiguous classes $C_{k,5}^{(sigma)}$ has rank $1$ or $2$.

We review some recent results on the phenomenon of surface superconductivity in the framework of Ginzburg-Landau theory for extreme type-II materials. In particular, we focus on the response of the superconductor to a strong longitudinal magnetic field in the regime where superconductivity survives only along the boundary of the wire. We derive the energy and density asymptotics for samples with smooth cross section, up to curvature-dependent terms. Furthermore, we discuss the corrections in presence of corners at the boundary of the sample.

We compute non-extremal three-point functions of scalar operators in $mathcal{N}=4$ super Yang-Mills at tree-level in $g_{YM}$ and at finite $N_c$, using the operator basis of the restricted Schur characters. We make use of the diagrammatic methods called quiver calculus to simplify the three-point functions. The results involve an invariant product of the generalized Racah-Wigner tensors ($6j$ symbols). Assuming that the invariant product is written by the Littlewood-Richardson coefficients, we show that the non-extremal three-point functions satisfy the large $N_c$ background independence; correspondence between the string excitations on $AdS_5 imes S^5$ and those in the LLM geometry.

This paper deals with an optimal control problem associated to the Kuramoto model describing the dynamical behavior of a network of coupled oscillators. Our aim is to design a suitable control function allowing us to steer the system to a synchronized configuration in which all the oscillators are aligned on the same phase. This control is computed via the minimization of a given cost functional associated with the dynamics considered. For this minimization, we propose a novel approach based on the combination of a standard Gradient Descent (GD) methodology with the recently-developed Random Batch Method (RBM) for the efficient numerical approximation of collective dynamics. Our simulations show that the employment of RBM improves the performances of the GD algorithm, reducing the computational complexity of the minimization process and allowing for a more efficient control calculation.

This paper considers the distributed nonconvex optimization problem of minimizing a global cost function formed by a sum of local cost functions by using local information exchange. We first propose a distributed first-order primal-dual algorithm. We show that it converges sublinearly to the stationary point if each local cost function is smooth and linearly to the global optimum under an additional condition that the global cost function satisfies the Polyak-{L}ojasiewicz condition. This condition is weaker than strong convexity, which is a standard condition for proving the linear convergence of distributed optimization algorithms, and the global minimizer is not necessarily unique or finite. Motivated by the situations where the gradients are unavailable, we then propose a distributed zeroth-order algorithm, derived from the proposed distributed first-order algorithm by using a deterministic gradient estimator, and show that it has the same convergence properties as the proposed first-order algorithm under the same conditions. The theoretical results are illustrated by numerical simulations.

We investigate quasistatic evolution in finite plasticity under the assumption that the plastic strain is compatible. This assumption is well-suited to describe the special case of dislocation-free plasticity and entails that the plastic strain is the gradient of a plastic deformation map. The total deformation can be then seen as the composition of a plastic and an elastic deformation. This opens the way to an existence theory for the quasistatic evolution problem featuring both Lagrangian and Eulerian variables. A remarkable trait of the result is that it does not require second-order gradients.

Yang-Mills gauge theory on Minkowski space supports a Batalin-Vilkovisky-infinity algebra structure, all whose operations are local. To make this work, the axioms for a BV-infinity algebra are deformed by a quadratic element, here the Minkowski wave operator. This homotopy structure implies BCJ/color-kinematics duality; a cobar construction yields a strict algebraic structure whose Feynman expansion for Yang-Mills tree amplitudes complies with the duality. It comes with a `syntactic kinematic algebra'.

We study the dependence of the first eigenvalue of the Finsler $p$-Laplacian and the corresponding eigenfunctions upon perturbation of the domain and we generalize a few results known for the standard $p$-Laplacian. In particular, we prove a Frech'{e}t differentiability result for the eigenvalues, we compute the corresponding Hadamard formulas and we prove a continuity result for the eigenfunctions. Finally, we briefly discuss a well-known overdetermined problem and we show how to deduce the Rellich-Pohozaev identity for the Finsler $p$-Laplacian from the Hadamard formula.

We first give a sufficient condition, issued from pluripotential theory, for an unbounded domain in the complex Euclidean space $mathbb C^n$ to be Kobayashi hyperbolic. Then, we construct an example of a rigid pseudoconvex domain in $mathbb C^3$ that is Kobayashi hyperbolic and has a nonempty core. In particular, this domain is not biholomorphic to a bounded domain in $mathbb C^3$ and the mentioned above sufficient condition for Kobayashi hyperbolicity is not necessary.

We show that the local equivalence class of the collapsed link Floer complex $cCFL^infty(L)$, together with many $Upsilon$-type invariants extracted from this group, is a concordance invariant of links. In particular, we define a version of the invariants $Upsilon_L(t)$ and $ u^+(L)$ when $L$ is a link and we prove that they give a lower bound for the slice genus $g_4(L)$. Furthermore, in the last section of the paper we study the homology group $HFL'(L)$ and its behaviour under unoriented cobordisms. We obtain that a normalized version of the $upsilon$-set, introduced by Ozsv'ath, Stipsicz and Szab'o, produces a lower bound for the 4-dimensional smooth crosscap number $gamma_4(L)$.

Motivated by quotient algorithms, such as the well-known $p$-quotient or solvable quotient algorithms, we describe how to compute extensions $ ilde H$ of a finite group $H$ by a direct sum of isomorphic simple $mathbb{Z}_p H$-modules such that $H$ and $ ilde H$ have the same number of generators. Similar to other quotient algorithms, our description will be via a suitable covering group of $H$. Defining this covering group requires a study of the representation module, as introduced by Gasch"utz in 1954. Our investigation involves so-called Fox derivatives (coming from free differential calculus) and, as a by-product, we prove that these can be naturally described via a wreath product construction. An important application of our results is that they can be used to compute, for a given epimorphism $G o H$ and simple $mathbb{Z}_p H$-module $V$, the largest quotient of $G$ that maps onto $H$ with kernel isomorphic to a direct sum of copies of $V$. For this we also provide a description of how to compute second cohomology groups for the (not necessarily solvable) group $H$, assuming a confluent rewriting system for $H$. To represent the corresponding group extensions on the computer, we introduce a new hybrid format that combines this rewriting system with the polycyclic presentation of the module.

This work has been motivated by recent papers that quantify the density of values of generic quadratic forms and other polynomials at integer points, in particular ones that use Rogers' second moment estimates. In this paper we establish such results in a very general framework. Namely, given any subhomogeneous function (a notion to be defined) $f: mathbb{R}^n o mathbb{R}$, we derive a necessary and sufficient condition on the approximating function $psi$ for guaranteeing that a generic element $fcirc g$ in the $G$-orbit of $f$ is $psi$-approximable; that is, $|fcirc g(mathbf{v})| le psi(|mathbf{v}|)$ for infinitely many $mathbf{v} in mathbb{Z}^n$. We also deduce a sufficient condition in the case of uniform approximation. Here, $G$ can be any closed subgroup of $operatorname{ASL}_n(mathbb{R})$ satisfying certain axioms that allow for the use of Rogers-type estimates.

Let $chi_{Delta}(mathbb{R}^{n})$ denote the minimum number of colors needed to color $mathbb{R}^{n}$ so that there will not be a monochromatic equilateral triangle with side length $1$. Using the slice rank method, we reprove a result of Frankl and Rodl, and show that $chi_{Delta}left(mathbb{R}^{n} ight)$ grows exponentially with $n$. This technique substantially improves upon the best known quantitative lower bounds for $chi_{Delta}left(mathbb{R}^{n} ight)$, and we obtain [ chi_{Delta}left(mathbb{R}^{n} ight)>(1.01446+o(1))^{n}. ]

In this work, we study dynamic properties of classical solutions to a homogenous Neumann initial-boundary value problem (IBVP) for a two-species and two-stimuli chemotaxis model with/without chemical signalling loop in a 2D bounded and smooth domain. We successfully detect the product of two species masses as a feature to determine boundedness, gradient estimates, blow-up and $W^{j,infty}(1leq jleq 3)$-exponential convergence of classical solutions for the corresponding IBVP. More specifically, we first show generally a smallness on the product of both species masses, thus allowing one species mass to be suitably large, is sufficient to guarantee global boundedness, higher order gradient estimates and $W^{j,infty}$-convergence with rates of convergence to constant equilibria; and then, in a special case, we detect a straight line of masses on which blow-up occurs for large product of masses. Our findings provide new understandings about the underlying model, and thus, improve and extend greatly the existing knowledge relevant to this model.

In a multitype branching process, it is assumed that immigrants arrive according to a nonhomogeneous Poisson or a generalized Polya process (both processes are formulated as a nonhomogeneous birth process with an appropriate choice of transition intensities). We show that the renormalized numbers of objects of the various types alive at time $t$ for supercritical, critical, and subcritical cases jointly converge in distribution under those two different arrival processes. Furthermore, some transient moment analysis when there are only two types of particles is provided. AMS 2000 subject classifications: Primary 60J80, 60J85; secondary 60K10, 60K25, 90B15.

We study the regularity of exceptional actions of groups by $C^{1,alpha}$ diffeomorphisms on the circle, i.e. ones which admit exceptional minimal sets, and whose elements have first derivatives that are continuous with concave modulus of continuity $alpha$. Let $G$ be a finitely generated group admitting a $C^{1,alpha}$ action $ ho$ with a free orbit on the circle, and such that the logarithms of derivatives of group elements are uniformly bounded at some point of the circle. We prove that if $G$ has spherical growth bounded by $c n^{d-1}$ and if the function $1/alpha^d$ is integrable near zero, then under some mild technical assumptions on $alpha$, there is a sequence of exceptional $C^{1,alpha}$ actions of $G$ which converge to $ ho$ in the $C^1$ topology. As a consequence for a single diffeomorphism, we obtain that if the function $1/alpha$ is integrable near zero, then there exists a $C^{1,alpha}$ exceptional diffeomorphism of the circle. This corollary accounts for all previously known moduli of continuity for derivatives of exceptional diffeomorphisms. We also obtain a partial converse to our main result. For finitely generated free abelian groups, the existence of an exceptional action, together with some natural hypotheses on the derivatives of group elements, puts integrability restrictions on the modulus $alpha$. These results are related to a long-standing question of D. McDuff concerning the length spectrum of exceptional $C^1$ diffeomorphisms of the circle.

In this paper, we provide some topological criteria for the Poisson Dixmier-Moeglin equivalence for $A$ in terms of the poset $({ m P. spec A}, subseteq)$ and the symplectic leaf or core stratification on its maximal spectrum. In particular, we prove that the Zariski topology of the Poisson prime spectrum and of each symplectic leaf or core can detect the Poisson Dixmier-Moeglin equivalence for any complex affine Poisson algebra. Moreover, we generalize the weaker version of the Poisson Dixmier-Moeglin equivalence for a complex affine Poisson algebra proved in [J. Bell, S. Launois, O.L. S'anchez, and B. Moosa, Poisson algebras via model theory and differential algebraic geometry, J. Eur. Math. Soc. (JEMS), 19(2017), no. 7, 2019-2049] to the general context of a commutative differential algebra.

The goal of this article is to investigate infinite dimensional affine diffusion processes on the canonical state space. This includes a derivation of the corresponding system of Riccati differential equations and an existence proof for such processes, which has been missing in the literature so far. For the existence proof, we will regard affine processes as solutions to infinite dimensional stochastic differential equations with values in Hilbert spaces. This requires a suitable version of the Yamada-Watanabe theorem, which we will provide in this paper. Several examples of infinite dimensional affine processes accompany our results.

We study an equivariant extension of the Batalin-Vilkovisky formalism for quantizing gauge theories. Namely, we introduce a general framework to encompass failures of the quantum master equation, and we apply it to the natural equivariant extension of AKSZ solutions of the classical master equation (CME). As examples of the construction, we recover the equivariant extension of supersymmetric Yang-Mills in 2d and of Donaldson-Witten theory.

This paper is devoted to the study of the singularity phenomenon of timelike extremal hypersurfaces in Minkowski spacetime $mathbb{R}^{1+3}$. We find that there are two explicit lightlike self-similar solutions to a graph representation of timelike extremal hypersurfaces in Minkowski spacetime $mathbb{R}^{1+3}$, the geometry of them are two spheres. The linear mode unstable of those lightlike self-similar solutions for the radially symmetric membranes equation is given. After that, we show those self-similar solutions of the radially symmetric membranes equation are nonlinearly stable inside a strictly proper subset of the backward lightcone. This means that the dynamical behavior of those two spheres is as attractors. Meanwhile, we overcome the double roots case (the theorem of Poincar'{e} can't be used) in solving the difference equation by construction of a Newton's polygon when we carry out the analysis of spectrum for the linear operator.

We introduce an infinite-dimensional $p$-adic affine group and construct its irreducible unitary representation. Our approach follows the one used by Vershik, Gelfand and Graev for the diffeomorphism group, but with modifications made necessary by the fact that the group does not act on the phase space. However it is possible to define its action on some classes of functions.

We introduce primal and dual stochastic gradient oracle methods for decentralized convex optimization problems. Both for primal and dual oracles the proposed methods are optimal in terms of the number of communication steps. However, for all classes of the objective, the optimality in terms of the number of oracle calls per node in the class of methods with optimal number of communication steps takes place only up to a logarithmic factor and the notion of smoothness. By using mini-batching technique we show that all proposed methods with stochastic oracle can be additionally parallelized at each node.

The central aim of this paper is to study (regional) fractional Poincar'e type inequalities on unbounded domains satisfying the finite ball condition. Both existence and non existence type results are established depending on various conditions on domains and on the range of $s in (0,1)$. The best constant in both regional fractional and fractional Poincar'e inequality is characterized for strip like domains $(omega imes mathbb{R}^{n-1})$, and the results obtained in this direction are analogous to those of the local case. This settles one of the natural questions raised by K. Yeressian in [ extit{Asymptotic behavior of elliptic nonlocal equations set in cylinders, Asymptot. Anal. 89, (2014), no 1-2}].

We prove that, given a constant $K> 2$ and a bounded linear operator $T$ from a JB$^*$-triple $E$ into a complex Hilbert space $H$, there exists a norm-one functional $psiin E^*$ satisfying $$|T(x)| leq K , |T| , |x|_{psi},$$ for all $xin E$. Applying this result we show that, given $G > 8 (1+2sqrt{3})$ and a bounded bilinear form $V$ on the Cartesian product of two JB$^*$-triples $E$ and $B$, there exist norm-one functionals $varphiin E^{*}$ and $psiin B^{*}$ satisfying $$|V(x,y)| leq G |V| , |x|_{varphi} , |y|_{psi}$$ for all $(x,y)in E imes B$. These results prove a conjecture pursued during almost twenty years.

The Gabriel-Roiter measure is used to give an alternative proof of the finiteness of the representation dimension for Artin algebras, a result established by Iyama in 2002. The concept of Gabriel-Roiter measure can be extended to abelian length categories and every such category has multiple Gabriel-Roiter measures. Using this notion, we prove the following broader statement: given any object $X$ and any Gabriel-Roiter measure $mu$ in an abelian length category $mathcal{A}$, there exists an object $X'$ which depends on $X$ and $mu$, such that $Gamma = operatorname{End}_{mathcal{A}}(X oplus X')$ has finite global dimension. Analogously to Iyama's original results, our construction yields quasihereditary rings and fits into the theory of rejective chains.

We deal with various Diophantine equations involving the Euler totient function and various sequences of numbers, including factorials, powers, and Fibonacci sequences.

We consider Landau-Ginzburg models stemming from groups comprised of non-diagonal symmetries, and we describe a rule for the mirror LG model. In particular, we present the non-abelian dual group, which serves as the appropriate choice of group for the mirror LG model. We also describe an explicit mirror map between the A-model and the B-model state spaces for two examples. Further, we prove that this mirror map is an isomorphism between the untwisted broad sectors and the narrow diagonal sectors for Fermat type polynomials.

In this paper we study the following set[A={p(n)+2^nd mod 1: ngeq 1}subset [0.1],] where $p$ is a polynomial with at least one irrational coefficient on non constant terms, $d$ is any real number and for $ain [0,infty)$, $a mod 1$ is the fractional part of $a$. By a Bernoulli decomposition method, we show that the closure of $A$ must have full Hausdorff dimension.

We study the geometry and topology of exotic Springer fibers for orbits corresponding to one-row bipartitions from an explicit, combinatorial point of view. This includes a detailed analysis of the structure of the irreducible components and their intersections as well as the construction of an explicit affine paving. Moreover, we compute the ring structure of cohomology by constructing a CW-complex homotopy equivalent to the exotic Springer fiber. This homotopy equivalent space admits an action of the type C Weyl group inducing Kato's original exotic Springer representation on cohomology. Our results are described in terms of the diagrammatics of the one-boundary Temperley-Lieb algebra (also known as the blob algebra). This provides a first step in generalizing the geometric versions of Khovanov's arc algebra to the exotic setting.

We derive finite rational formulas for the traces of cycle integrals of certain meromorphic modular forms. Moreover, we prove the modularity of a completion of the generating function of such traces. The theoretical framework for these results is an extension of the Shintani theta lift to meromorphic modular forms of positive even weight.