<Emphasis Type="Italic">O</Emphasis>(1) loop model with different boundary conditions and symmetry classes of alternating-sign matrices

This work is a continuation of our recent paper where we discussed numerical evidence that the numbers of the states of the fully packed loop model with fixed pairing patterns coincide with the components of the ground state vector of the O(1) loop model with periodic boundary conditions and an even number of sites. We give two new conjectures related to different boundary conditions: we suggest and numerically verify that the numbers of the half-turn symmetric states of the fully packed loop model with fixed pairing patterns coincide with the components of the ground state vector of the O(1) loop model with periodic boundary conditions and an odd number of sites and that the corresponding numbers of the vertically symmetric states describe the case of open boundary conditions and an even number of sites.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 142, No. 2, pp. 284–292, February, 2005.     

Existence of the non-radially symmetric ground state for p-Laplacian equations involving Choquard type

We consider some p-Laplacian type equations with sum of nonlocal term and subcritical nonlinearities. We prove the existence of the ground states, which are positive. Because of including p=2, these results extend the results of Li, Ma and Zhang [Nonlinear Analysis: Real World Application 45(2019) 1-25]. When p=2, N=3, by a variant variational identity and a constraint set, we can prove the existence of a non-radially symmetric solution. Moreover, this solution u(x₁, x₂, x₃) is radially symmetric with respect to (x₁, x₂) and odd with respect to x₃.     

Gell-Mann and Low Formula for Degenerate Unperturbed States

The Gell-Mann and Low switching allows to transform eigenstates of an unperturbed Hamiltonian H ₀ into eigenstates of the modified Hamiltonian H ₀ + V. This switching can be performed when the initial eigenstate is not degenerate, under some gap conditions with the remainder of the spectrum. We show here how to extend this approach to the case when the ground state of the unperturbed Hamiltonian is degenerate. More precisely, we prove that the switching procedure can still be performed when the initial states are eigenstates of the finite rank self-adjoint operator P₀VP₀{\mathcal{P}_{0}V\mathcal{P}_{0}} , where P₀{\mathcal{P}_0} is the projection onto a degenerate eigenspace of H ₀.     

On solutions of a system of hereditary and self-referred partial-differential equations

In this paper, we present the local and global solutions of a system of hereditary and self-referred partial-differential equations. Namely, by the assumption on the Lipschitz continuity of the initial conditions u ₀, v ₀, Theorem 1 states the existence of local solutions of the problem (1.3–1.4); furthermore, under the assumption that those initial conditions are non-negative, non-decreasing, bounded, and lower semi-continuous functions, Theorem 2 gives global solution which is also a non-negative, non-decreasing, bounded, and lower semi-continuous function (in variable x of even for any time t).     

On Stable Self-Similar Blow up for Equivariant Wave Maps: The Linearized Problem

We consider co-rotational wave maps from (3 + 1) Minkowski space into the three-sphere. This is an energy supercritical model which is known to exhibit finite time blow up via self-similar solutions. The ground state self-similar solution f ₀ is known in closed form and based on numerics, it is supposed to describe the generic blow up behavior of the system. In this paper we develop a rigorous linear perturbation theory around f ₀. This is an indispensable prerequisite for the study of nonlinear stability of the self-similar blow up which is conducted in the companion paper (Donninger in Commun. Pure Appl. Math., 64(8), 2011). In particular, we prove that f ₀ is linearly stable if it is mode stable. Furthermore, concerning the mode stability problem, we prove new results that exclude the existence of unstable eigenvalues with large imaginary parts and also, with real parts larger than \frac12{\frac{1}{2}}. The remaining compact region is well-studied numerically and all available results strongly suggest the nonexistence of unstable modes.     

Asymptotic estimates and stability analysis of Kuramoto-Sivashinsky type models

We first show asymptotic L ² bounds for a class of equations, which includes the Burger-Sivashinsly model for odd solutions with periodic boundary conditions. We consider the conditional stability of stationary solutions of Kuramoto-Sivashinsky equation in the periodic setting. We establish rigorously the general structure of the spectrum of the linearized operator, in particular the linear instability of steady states. In addition, we show conditional asymptotic stability with asymptotic phase, under a natural spectral hypothesis for the corresponding linearized operator. For the zero solution, we have more precise results. Namely, in the non-resonant regime L ≠ n π, we prove conditional asymptotic stability, provided one considers only mean value zero data. If, however, L = n ₀ π (but still ò\nolimits_-L^L u₀(x) dx=0{\int\nolimits_{-L}^L u_0(x) dx=0}), then we have conditional orbital stability. More specifically, the solutions relax to a small (but generally non-zero) function as long as the initial data are small and lie on a center-stable manifold of codimension 2(n ₀ − 1).     

The automorphism groups of the vertex operator algebras V + L : general case

In this article, we give a method of calculating the automorphism groups of the vertex operator algebras V⁺_L associated with even lattices L. For example, by using this method we determine the automorphism groups of V⁺_L for even lattices of rank one, two and three, and even unimodular lattices. The author was supported by the Japan Society for the Promotion of Science Research Fellowships for Young Scientists and COE grant of Hokkaido University.     

On stable self‐similar blowup for equivariant wave maps

We consider corotational wave maps from (3 + 1) Minkowski space into the 3‐sphere. This is an energy supercritical model that is known to exhibit finite‐time blowup via self‐similar solutions. The ground state self‐similar solution f₀ is known in closed form, and according to numerics, it describes the generic blowup behavior of the system. We prove that the blowup via f₀ is stable under the assumption that f₀ does not have unstable modes. This condition is equivalent to a spectral assumption for a linear second order ordinary differential operator. In other words, we reduce the problem of stable blowup to a linear ODE spectral problem. Although we are unable at the moment to verify the mode stability of f₀ rigorously, it is known that possible unstable eigenvalues are confined to a certain compact region in the complex plane. As a consequence, highly reliable numerical techniques can be applied and all available results strongly suggest the nonexistence of unstable modes, i.e., the assumed mode stability of f₀. © 2011 Wiley Periodicals, Inc.     

Qualitative analysis of the SICR epidemic model with impulsive vaccinations

Control of epidemic infections is a very urgent issue today. To develop an appropriate strategy for vaccinations and effectively prevent the disease from arising and spreading, we proposed a modified Susceptible‐Infected‐Removed model with impulsive vaccinations. For the model without vaccinations, we proved global stability of one of the steady states depending on the basic reproduction number R₀. As typically in the epidemic models, the threshold value of R₀ is 1. If R₀ is greater than 1, then the positive steady state called endemic equilibrium exists and is globally stable, whereas for smaller values of R₀, it does not exist, and the semi‐trivial steady state called disease‐free equilibrium is globally stable. Using impulsive differential equation comparison theorem, we derived sufficient conditions under which the infectious disease described by the considered model disappears ultimately. The analytical results are illustrated by numerical simulations for Hepatitis B virus infection that confirm the theoretical possibility of the infection elimination because of the proper vaccinations policy. Copyright © 2012 John Wiley & Sons, Ltd.     

Three results on the regularity of the curves that are invariant by an exact symplectic twist map

A theorem due to G. D. Birkhoff states that every essential curve which is invariant under a symplectic twist map of the annulus is the graph of a Lipschitz map. We prove: if the graph of a Lipschitz map h:T→R is invariant under a symplectic twist map, then h is a little bit more regular than simply Lipschitz (Theorem 1); we deduce that there exists a Lipschitz map h:T→R whose graph is invariant under no symplectic twist map (Corollary 2). Assuming that the dynamic of a twist map restricted to a Lipschitz graph is bi-Lipschitz conjugate to a rotation, we obtain that the graph is even C ¹ (Theorem 3). Then we consider the case of the C ⁰ integrable symplectic twist maps and we prove that for such a map, there exists a dense G _δ subset of the set of its invariant curves such that every curve of this G _δ subset is C ¹ (Theorem 4).     

Local and global robustness at steady state

We study the robustness of the steady states of a class of systems of autonomous ordinary differential equations (ODEs), having as a central example those arising from (bio)chemical reaction networks. More precisely, we study under what conditions the steady states of the system are contained in a parallel translate of a coordinate hyperplane. To this end, we focus mainly on ODEs consisting of generalized polynomials and make use of algebraic and geometric tools to relate the local and global structure of the set of steady states. Specifically, we consider the local property termed zero sensitivity at a coordinate x_i, which means that the tangent space is contained in a hyperplane of the form x_i = c, and provide a criterion to identify it. We consider the global property termed absolute concentration robustness (ACR), meaning that all steady states are contained in a hyperplane of the form x_i = c. We clarify and formalize the relation between the two approaches. In particular, we show that ACR implies zero sensitivity and identify when the two properties do not agree, via an intermediate property we term local ACR. For families of systems arising from modeling biochemical reaction networks, we obtain the first practical and automated criterion to decide upon (local) ACR.     

Homological stability for configuration spaces of manifolds

Let C _n (M) be the configuration space of n distinct ordered points in M. We prove that if M is any connected orientable manifold (closed or open), the homology groups H _i (C _n (M);ℚ) are representation stable in the sense of Church and Farb (). Applying this to the trivial representation, we obtain as a corollary that the unordered configuration space B _n (M) satisfies classical homological stability: for each i, H _i (B _n (M);ℚ)≈H _i (B _n+1(M);ℚ) for n>i. This improves on results of McDuff, Segal, and others for open manifolds. Applied to closed manifolds, this provides natural examples where rational homological stability holds even though integral homological stability fails.     

Zero level of a purely magnetic two-dimensional nonrelativistic Pauli operator for SPIN-1/2 particles

We study the manifold of complex Bloch-Floquet eigenfunctions for the zero level of a two-dimensional nonrelativistic Pauli operator describing the propagation of a charged particle in a periodic magnetic field with zero flux through the elementary cell and a zero electric field. We study this manifold in full detail for a wide class of algebraic-geometric operators. In the nonzero flux case, the Pauli operator ground state was found by Aharonov and Casher for fields rapidly decreasing at infinity and by Dubrovin and Novikov for periodic fields. Algebraic-geometric operators were not previously known for fields with nonzero flux because the complex continuation of “magnetic” Bloch-Floquet eigenfunctions behaves wildly at infinity. We construct several nonsingular algebraic-geometric periodic fields (with zero flux through the elementary cell) corresponding to complex Riemann surfaces of genus zero. For higher genera, we construct periodic operators with interesting magnetic fields and with the Aharonov-Bohm phenomenon. Algebraic-geometric solutions of genus zero also generate soliton-like nonsingular magnetic fields whose flux through a disc of radius R is proportional to R (and diverges slowly as R → ∞). In this case, we find the most interesting ground states in the Hilbert space L ₂ (ℝ ² ).     

Weakly periodic ground states and Gibbs measures for the Ising model with competing interactions on the Cayley tree

We introduce the notion of a weakly periodic configuration. For the Ising model with competing interactions, we describe the set of all weakly periodic ground states corresponding to normal divisors of indices 2 and 4 of the group representation of the Cayley tree. In addition, we study new Gibbs measures for the Ising model.     

The Ground State and the Long-Time Evolution in the CMC Einstein Flow

Let (g, K)(k) be a CMC (vacuum) Einstein flow over a compact three-manifold Σ with non-positive Yamabe invariant (Y(Σ)). As noted by Fischer and Moncrief, the reduced volume ${\mathcal{V}(k)=\left(\frac{-k}{3}\right)^{3}{\rm Vol}_{g(k)}(\Sigma)}Let (g, K)(k) be a CMC (vacuum) Einstein flow over a compact three-manifold Σ with non-positive Yamabe invariant (Y(Σ)). As noted by Fischer and Moncrief, the reduced volume V(k)=(\frac-k3)³Vol_g(k)(S){\mathcal{V}(k)=\left(\frac{-k}{3}\right)^{3}{\rm Vol}_{g(k)}(\Sigma)} is monotonically decreasing in the expanding direction and bounded below by V_inf=(\frac-16Y(S))^\frac32{\mathcal{V}_{\rm \inf}=\left(\frac{-1}{6}Y(\Sigma)\right)^{\frac{3}{2}}}. Inspired by this fact we define the ground state of the manifold Σ as “the limit” of any sequence of CMC states {(g _i , K _i )} satisfying: (i) k _i  = −3, (ii) V_iˉ V_inf{\mathcal{V}_{i}\downarrow \mathcal{V}_{\rm inf}}, (iii) Q ₀((g _i , K _i )) ≤ Λ, where Q ₀ is the Bel–Robinson energy and Λ is any arbitrary positive constant. We prove that (as a geometric state) the ground state is equivalent to the Thurston geometrization of Σ. Ground states classify naturally into three types. We provide examples for each class, including a new ground state (the Double Cusp) that we analyze in detail. Finally, consider a long time and cosmologically normalized flow ([(g)\tilde],[(K)\tilde])(s)=((\frac-k3)²g,(\frac-k3)K){(\tilde{g},\tilde{K})(\sigma)=\left(\left(\frac{-k}{3}\right)^{2}g,\left(\frac{-k}{3}\right)K\right)}, where s = -ln(-k) ? [a,￥){\sigma=-\ln (-k)\in [a,\infty)}. We prove that if [(E₁)\tilde]=E₁(([(g)\tilde],[(K)\tilde])) ￡ L{\tilde{\mathcal{E}_{1}}=\mathcal{E}_{1}((\tilde{g},\tilde{K}))\leq \Lambda} (where E₁=Q₀+Q₁{\mathcal{E}_{1}=Q_{0}+Q_{1}}, is the sum of the zero and first order Bel–Robinson energies) the flow ([(g)\tilde],[(K)\tilde])(s){(\tilde{g},\tilde{K})(\sigma)} persistently geometrizes the three-manifold Σ and the geometrization is the ground state if Vˉ V_inf{\mathcal{V}\downarrow \mathcal{V}_{\rm inf}}.     

On the Weak-Integrity of Graphs

Connectivity has been used in the past to describe the stability of graphs. If two graphs have the same connectivity, then it does not distinguish between these graphs. That is, the connectivity is not a good measure of graph stability. Then we need other graph parameters to describe the stability. Suppose that two graphs have the same connectivity and the order (the number of vertices or edges) of the largest components of these graphs are not equal. Hence, we say that these graphs must be different in respect to stability and so we can define a new measure which distinguishes these graphs. In this paper, the Weak-Integrity of a graph G is introduced as a new measure of stability in this sense and it is defined as I _w (G)=min _SV(G){S+m _e (G–S)}, where m _e (G–S) denotes the number of edges of the largest component of G–S. We give the weak-integrity of graphs obtained via various operations that are unary, such as powers, and binary, such as union, composition, product and corona.     

Improved K1-stability for the embedding D5 into E6

Abstract

This article is dedicated to the surjective stability of the K₁-functor for Chevalley groups for the embedding D₅ into E₆. This case was already studied by Plotkin. In this article, we improve his result by showing that surjective stability holds under a weaker assumption on a ring. Another result of this article shows how the K₁-stability can help to study overgroups of subsystem subgroups.     

Stability of Gabor frames with arbitrary sampling points

Summary We study the stability of Gabor frames with arbitrary sampling points in the time-frequency plane, in several aspects. We prove that a Gabor frame generated by a window function in the Segal algebra S₀(R^d) remains a frame even if (possibly) all the sampling points undergo an arbitrary perturbation, as long as this is uniformly small. We give explicit stability bounds when the window function is nice enough, showing that the allowed perturbation depends only on the lower frame bound of the original family and some qualitative parameters of the window under consideration. For the perturbation of window functions we show that a Gabor frame generated by any window function with arbitrary sampling points remains a frame when the window function has a small perturbation in S₀(R^d) sense. We also study the stability of dual frames, which is useful in practice but has not found much attention in the literature. We give some general results on this topic and explain consequences to Gabor frames.     

On the decay rates of Timoshenko system with second sound

In this work, we study the well‐posedness and the asymptotic stability of a one‐dimensional linear thermoelastic Timoshenko system, where the heat conduction is given by Cattaneo's law and the coupling is via the displacement equation. We prove that the system is exponentially stable provided that the stability number χ_τ=0. Otherwise, we show that the system lacks exponential stability. Furthermore, in the latter case, we show that the solution decays polynomially. Copyright © 2015 John Wiley & Sons, Ltd.     

Stability of the Quadratic Equation of Pexider Type

We will investigate the stability problem of the quadratic equation (1) and extend the results of Borelli and Forti, Czerwik, and Rassias. By applying this result and an improved theorem of the author, we will also prove the stability of the quadratic functional equation of Pexider type,f ₁ (x +y) + f₂(x -y) =f ₃(x) +f ₄(y), for a large class of functions.