E α -Ulam type stability of fractional order ordinary differential equations

In this paper, the concepts of $\mathbb{E}_{\alpha}$ -Ulam-Hyers stability, generalized $\mathbb{E}_{\alpha}$ -Ulam-Hyers stability, $\mathbb{E}_{\alpha}$ -Ulam-Hyers-Rassias stability and generalized $\mathbb{E}_{\alpha}$ -Ulam-Hyers-Rassias stability for fractional order ordinary differential equations are raised. Without loss of generality, $\mathbb{E}_{\alpha}$ -Ulam-Hyers-Rassias stability result is derived by using a singular integral inequality of Gronwall type. Two examples are also provided to illustrate our results.     

Norm-Constrained Determinantal Representations of Multivariable Polynomials

For every multivariable polynomial $p$ , with $p(0)=1$ , we construct a determinantal representation, $ p=\det (I - K Z )$ , where $Z$ is a diagonal matrix with coordinate variables on the diagonal and $K$ is a complex square matrix. Such a representation is equivalent to the existence of $K$ whose principal minors satisfy certain linear relations. When norm constraints on $K$ are imposed, we give connections to the multivariable von Neumann inequality, Agler denominators, and stability. We show that if a multivariable polynomial $q$ , $q(0)=0,$ satisfies the von Neumann inequality, then $1-q$ admits a determinantal representation with $K$ a contraction. On the other hand, every determinantal representation with a contractive $K$ gives rise to a rational inner function in the Schur–Agler class.     

Non-Archimedean stability of a quadratic functional equation

Recently the Hyers–Ulam stability of the quadratic functional equation $ f(kx+y)+f(kx+\sigma (y))- 2k^2 f(x)-2f(y)=0 $ where $\sigma $ is an involution of the normed space $E$ and $k$ is a fixed positive integer greater that 1, has been proved in the earlier work. In this paper, using fixed point and direct methods, we prove the Hyers–Ulam stability of the above functional equation in non-Archimedean normed spaces.     

Logarithmic Expansions and the Stability of Periodic Patterns of Localized Spots for Reaction–Diffusion Systems in {\mathbb {R}}^2

The linear stability of steady-state periodic patterns of localized spots in \({\mathbb {R}}^2\) for the two-component Gierer–Meinhardt (GM) and Schnakenberg reaction–diffusion models is analyzed in the semi-strong interaction limit corresponding to an asymptotically small diffusion coefficient \({\displaystyle \varepsilon }^2\) of the activator concentration. In the limit \({\displaystyle \varepsilon }\rightarrow 0\) , localized spots in the activator are centered at the lattice points of a Bravais lattice with constant area \(|\Omega |\) . To leading order in \(\nu ={-1/\log {\displaystyle \varepsilon }}\) , the linearization of the steady-state periodic spot pattern has a zero eigenvalue when the inhibitor diffusivity satisfies \(D={D_0/\nu }\) for some \(D_0\) independent of the lattice and the Bloch wavevector \({\pmb k}\) . From a combination of the method of matched asymptotic expansions, Floquet–Bloch theory, and the rigorous study of certain nonlocal eigenvalue problems, an explicit analytical formula for the continuous band of spectrum that lies within an \({\mathcal O}(\nu )\) neighborhood of the origin in the spectral plane is derived when \(D={D_0/\nu } + D_1\) , where \(D_1={\mathcal O}(1)\) is a detuning parameter. The periodic pattern is linearly stable when \(D_1\) is chosen small enough so that this continuous band is in the stable left half-plane \(\text{ Re }(\lambda )<0\) for all \({\pmb k}\) . Moreover, for both the Schnakenberg and GM models, our analysis identifies a model-dependent objective function, involving the regular part of the Bloch Green’s function, that must be maximized in order to determine the specific periodic arrangement of localized spots that constitutes a linearly stable steady-state pattern for the largest value of \(D\) . From a numerical computation, based on an Ewald-type algorithm, of the regular part of the Bloch Green’s function that defines the objective function, it is shown within the class of oblique Bravais lattices that a regular hexagonal lattice arrangement of spots is optimal for maximizing the stability threshold in \(D\) .     

Duality and the topological filtration

Let $(M,J)$ be a Fano manifold which admits a Kähler-Einstein metric $g_{KE}$ (or a Kähler-Ricci soliton $g_{KS}$ ). Then we prove that Kähler-Ricci flow on $(M,J)$ converges to $g_{KE}$ (or $g_{KS}$ ) in $C^\infty $ in the sense of Kähler potentials modulo holomorphisms transformation as long as an initial Kähler metric of flow is very closed to $g_{KE}$ (or $g_{KS}$ ). The result improves Main Theorem in [14] in the sense of stability of Kähler-Ricci flow.     

On the stability of \varphi -uniform domains

We study two metrics, the quasihyperbolic metric and the distance ratio metric of a subdomain $G \subset {\mathbb R}^n$ . In the sequel, we investigate a class of domains, so called $\varphi $ -uniform domains, defined by the property that these two metrics are comparable with respect to a homeomorphism $\varphi $ from $[0,\infty )$ to itself. Finally, we discuss a number of stability properties of $\varphi $ -uniform domains. In particular, we show that the class of $\varphi $ -uniform domains is stable in the sense that removal of a geometric sequence of points from a $\varphi $ -uniform domain yields a $\varphi _1$ -uniform domain.     

Stable hypersurfaces as minima of the integral of an anisotropic mean curvature preserving a linear combination of area and volume

We deal with a compact hypersurface $M$ without boundary immersed in Euclidean space $R^{n+1}$ with the quotient of anisotropic mean curvatures $\frac{(r+1)C_{n}^{r+1}H^F_{r+1}}{a(k+1)C_{n}^{k+1}H^F_{k+1}-b}=constant$ , for real numbers $a$ and $b$ . Such a hypersurface is a critical point for the variational problem preserving a linear combination (with coefficientes $a$ and $b$ ) of the $(k,F)$ -area and the $(n + 1)$ -volume enclosed by $M$ . We show that $M$ is $(r,k,a,b)$ -stable if and only if, up to translations and homotheties, it is the Wulff shape of $F$ , under some assumptions on $a$ and $b$ proved to be sharp. For $a=0$ and $b=1$ , this gives the known $r$ -stability of the $r$ -area for volume preserving variations; if also $F\equiv 1$ it yields the stability studied by Alencar-do Carmo-Rosenberg and Barbosa-Colares. For $b=0$ we also prove a characterization of the Wulff shape as a critical point of the $(r,F)$ -area for variations preserving the $(k, F)$ -area, $0\le k<r<n$ , without the $r$ -stability hypothesis.     

A T(1)-Theorem for non-integral operators

Let $X$ be a space of homogeneous type and let $L$ be a sectorial operator with bounded holomorphic functional calculus on $L^2(X)$ . We assume that the semigroup $\{e^{-tL}\}_{t>0}$ satisfies Davies–Gaffney estimates. Associated with $L$ are certain approximations of the identity. We call an operator $T$ a non-integral operator if compositions involving $T$ and these approximations satisfy certain weighted norm estimates. The Davies–Gaffney and the weighted norm estimates are together a substitute for the usual kernel estimates on $T$ in Calderón–Zygmund theory. In this paper, we show, under the additional assumption that a vertical Littlewood–Paley–Stein square function associated with $L$ is bounded on $L^2(X)$ , that a non-integral operator $T$ is bounded on $L^2(X)$ if and only if $T(1) \in BMO_L(X)$ and $T^{*}(1) \in BMO_{L^{*}}(X)$ . Here, $BMO_L(X)$ and $BMO_{L^{*}}(X)$ denote the recently defined $BMO(X)$ spaces associated with $L$ that generalize the space $BMO(X)$ of John and Nirenberg. Generalizing a recent result due to F. Bernicot, we show a second version of a $T(1)$ -Theorem under weaker off-diagonal estimates, which gives a positive answer to a question raised by him. As an application, we prove $L^2(X)$ -boundedness of a paraproduct operator associated with $L$ . We moreover study criterions for a $T(b)$ -Theorem to be valid.     

Moduli spaces of hyperelliptic curves with A and D singularities

We introduce moduli spaces of quasi-admissible hyperelliptic covers with at worst A and D singularities. The stability conditions for these moduli spaces depend on two rational parameters describing allowable singularities. For the extreme values of the parameters, we obtain the stacks of stable limits of $A_n$ and $D_n$ singularities, and the quotients of the miniversal deformation spaces of these singularities by natural $\mathbb G _m$ -actions. We interpret the intermediate spaces as log canonical models of the stacks of stable limits of $A_n$ and $D_n$ singularities.     

Chromatic Gallai identities operating on Lovász number

If $G$ is a triangle-free graph, then two Gallai identities can be written as $\alpha (G)+\overline{\chi }(L(G))=|V(G)|=\alpha (L(G))+\overline{\chi }(G)$ , where $\alpha $ and $\overline{\chi }$ denote the stability number and the clique-partition number, and $L(G)$ is the line graph of  $G$ . We show that, surprisingly, both equalities can be preserved for any graph $G$ by deleting the edges of the line graph corresponding to simplicial pairs of adjacent arcs, according to any acyclic orientation of  $G$ . As a consequence, one obtains an operator $\Phi $ which associates to any graph parameter $\beta $ such that $\alpha (G) \le \beta (G) \le \overline{\chi }(G)$ for all graph $G$ , a graph parameter $\Phi _\beta $ such that $\alpha (G) \le \Phi _\beta (G) \le \overline{\chi }(G)$ for all graph $G$ . We prove that $\vartheta (G) \le \Phi _\vartheta (G)$ and that $\Phi _{\overline{\chi }_f}(G)\le \overline{\chi }_f(G)$ for all graph  $G$ , where $\vartheta $ is Lovász theta function and $\overline{\chi }_f$ is the fractional clique-partition number. Moreover, $\overline{\chi }_f(G) \le \Phi _\vartheta (G)$ for triangle-free $G$ . Comparing to the previous strengthenings $\Psi _\vartheta $ and $\vartheta ^{+ \triangle }$ of $\vartheta $ , numerical experiments show that $\Phi _\vartheta $ is a significant better lower bound for $\overline{\chi }$ than $\vartheta $ .     

Triangle-Free Geometric Intersection Graphs with Large Chromatic Number

Several classical constructions illustrate the fact that the chromatic number of a graph may be arbitrarily large compared to its clique number. However, until very recently no such construction was known for intersection graphs of geometric objects in the plane. We provide a general construction that for any arc-connected compact set $X$ X in $\mathbb{R }^2$ R 2 that is not an axis-aligned rectangle and for any positive integer $k$ k produces a family $\mathcal{F }$ F of sets, each obtained by an independent horizontal and vertical scaling and translation of $X$ X , such that no three sets in $\mathcal{F }$ F pairwise intersect and $\chi (\mathcal{F })>k$ χ ( F ) > k . This provides a negative answer to a question of Gyárfás and Lehel for L-shapes. With extra conditions we also show how to construct a triangle-free family of homothetic (uniformly scaled) copies of a set with arbitrarily large chromatic number. This applies to many common shapes, like circles, square boundaries or equilateral L-shapes. Additionally, we reveal a surprising connection between coloring geometric objects in the plane and on-line coloring of intervals on the line.     

Some totally geodesic submanifolds of the nonlinear Grassmannian of a compact symmetric space

Let \(M\) and \(N\) be two connected smooth manifolds, where \(M\) is compact and oriented and \(N\) is Riemannian. Let \(\mathcal {E}\) be the Fréchet manifold of all embeddings of \(M\) in \(N\) , endowed with the canonical weak Riemannian metric. Let \(\sim \) be the equivalence relation on \(\mathcal {E}\) defined by \(f\sim g\) if and only if \(f=g\circ \phi \) for some orientation preserving diffeomorphism \(\phi \) of \(M\) . The Fréchet manifold \(\mathcal {S}= \mathcal {E}/_{\sim }\) of equivalence classes, which may be thought of as the set of submanifolds of \(N\) diffeomorphic to \(M\) and is called the nonlinear Grassmannian (or Chow manifold) of \(N\) of type \(M\) , inherits from \( \mathcal {E}\) a weak Riemannian structure. We consider the following particular case: \(N\) is a compact irreducible symmetric space and \(M\) is a reflective submanifold of \(N\) (that is, a connected component of the set of fixed points of an involutive isometry of \( N\) ). Let \(\mathcal {C}\) be the set of submanifolds of \(N\) which are congruent to \(M\) . We prove that the natural inclusion of \(\mathcal {C}\) in \(\mathcal {S}\) is totally geodesic.     

Uncertainty principles and sum complexes

Let \(p\) be a prime and let \(A\) be a nonempty subset of the cyclic group \(C_p\) . For a field \({\mathbb F}\) and an element \(f\) in the group algebra \({\mathbb F}[C_p]\) let \(T_f\) be the endomorphism of \({\mathbb F}[C_p]\) given by \(T_f(g)=fg\) . The uncertainty number \(u_{{\mathbb F}}(A)\) is the minimal rank of \(T_f\) over all nonzero \(f \in {\mathbb F}[C_p]\) such that \(\mathrm{supp}(f) \subset A\) . The following topological characterization of uncertainty numbers is established. For \(1 \le k \le p\) define the sum complex \(X_{A,k}\) as the \((k-1)\) -dimensional complex on the vertex set \(C_p\) with a full \((k-2)\) -skeleton whose \((k-1)\) -faces are all \(\sigma \subset C_p\) such that \(|\sigma |=k\) and \(\prod _{x \in \sigma }x \in A\) . It is shown that if \({\mathbb F}\) is algebraically closed then $$\begin{aligned} u_{{\mathbb F}}(A)=p-\max \{k :\tilde{H}_{k-1}(X_{A,k};{\mathbb F}) \ne 0\}. \end{aligned}$$ The main ingredient in the proof is the determination of the homology groups of \(X_{A,k}\) with field coefficients. In particular it is shown that if \(|A| \le k\) then \(\tilde{H}_{k-1}(X_{A,k};{\mathbb F}_p)\!=\!0.\)      

Principal graph stability and the jellyfish algorithm

We show that if the principal graph of a subfactor planar algebra of modulus $\delta >2$ is stable for two depths, then it must end in $A_{ finite }$ tails. This result is analogous to Popa’s theorem on principal graph stability. We use these theorems to show that an $n-1$ supertransitive subfactor planar algebra has jellyfish generators at depth $n$ if and only if its principal graph is a spoke graph. This is the published version of arxiv:1208.1564.     

Integer-valued polynomials over matrices and divided differences

Let $D$ be an integrally closed domain with quotient field $K$ and $n$ a positive integer. We give a characterization of the polynomials in $K[X]$ which are integer-valued over the set of matrices $M_n(D)$ in terms of their divided differences. A necessary and sufficient condition on $f\in K[X]$ to be integer-valued over $M_n(D)$ is that, for each $k$ less than $n$ , the $k$ th divided difference of $f$ is integral-valued on every subset of the roots of any monic polynomial over $D$ of degree $n$ . If in addition $D$ has zero Jacobson radical then it is sufficient to check the above conditions on subsets of the roots of monic irreducible polynomials of degree $n$ , that is, conjugate integral elements of degree $n$ over $D$ .     

The average size of Ramanujan sums over quadratic number fields

This article is concerned with Ramanujan sums ${c_{\mathcal{I}_1}(\mathcal{I}),}$ where ${\mathcal{I},\mathcal{I}_1}$ are integral ideals in an arbitrary quadratic number field ${\mathbb{Q}(\sqrt{d}).}$ In particular, the asymptotic behavior of sums of ${c_{\mathcal{I}_1}(\mathcal{I}),}$ over both ${\mathcal{I}}$ and ${c_{\mathcal{I}_1}(\mathcal{I}),}$ is investigated.     

A convergent convex splitting scheme for the periodic nonlocal Cahn-Hilliard equation

In this paper we devise a first-order-in-time, second-order-in-space, convex splitting scheme for the periodic nonlocal Cahn-Hilliard equation. The unconditional unique solvability, energy stability and \(\ell ^\infty (0, T; \ell ^4)\) stability of the scheme are established. Using the a-priori stabilities, we prove error estimates for our scheme, in both the \(\ell ^\infty (0, T; \ell ^2)\) and \(\ell ^\infty (0, T; \ell ^\infty )\) norms. The proofs of these estimates are notable for the fact that they do not require point-wise boundedness of the numerical solution, nor a global Lipschitz assumption or cut-off for the nonlinear term. The \(\ell ^2\) convergence proof requires no refinement path constraint, while the one involving the \(\ell ^\infty \) norm requires only a mild linear refinement constraint, \(s \le C h\) . The key estimates for the error analyses take full advantage of the unconditional \(\ell ^\infty (0, T; \ell ^4)\) stability of the numerical solution and an interpolation estimate of the form \(\left\| \phi \right\| _4 \le C \left\| \phi \right\| _2^\alpha \left\| \nabla _h\phi \right\| _2^{1-\alpha },\alpha = \frac{4-D}{4},D=1,2,3\) , which we establish for finite difference functions. We conclude the paper with some numerical tests that confirm our theoretical predictions.     

Neighbour-transitive codes in Johnson graphs

The Johnson graph \(J(v,k)\) has, as vertices, the \(k\) -subsets of a \(v\) -set \(\mathcal {V}\) and as edges the pairs of \(k\) -subsets with intersection of size \(k-1\) . We introduce the notion of a neighbour-transitive code in \(J(v,k)\) . This is a proper vertex subset \(\Gamma \) such that the subgroup \(G\) of graph automorphisms leaving \(\Gamma \) invariant is transitive on both the set \(\Gamma \) of ‘codewords’ and also the set of ‘neighbours’ of \(\Gamma \) , which are the non-codewords joined by an edge to some codeword. We classify all examples where the group \(G\) is a subgroup of the symmetric group \(\mathrm{Sym}\,(\mathcal {V})\) and is intransitive or imprimitive on the underlying \(v\) -set \(\mathcal {V}\) . In the remaining case where \(G\le \mathrm{Sym}\,(\mathcal {V})\) and \(G\) is primitive on \(\mathcal {V}\) , we prove that, provided distinct codewords are at distance at least \(3\) , then \(G\) is \(2\) -transitive on \(\mathcal {V}\) . We examine many of the infinite families of finite \(2\) -transitive permutation groups and construct surprisingly rich families of examples of neighbour-transitive codes. A major unresolved case remains.     

Stability of contact discontinuity for steady Euler system in infinite duct

In this paper, we prove stability of contact discontinuities for full Euler system. We fix a flat duct ${\mathcal{N}_0}$ of infinite length in ${\mathbb{R}^2}$ with width W ₀ and consider two uniform subsonic flow ${{U_l}^{\pm}=(u_l^{\pm}, 0, pl,\rho_l^{\pm})}$ with different horizontal velocity in ${\mathcal{N}_0}$ divided by a flat contact discontinuity ${\Gamma_{cd}}$ . And, we slightly perturb the boundary of ${\mathcal{N}_0}$ so that the width of the perturbed duct converges to ${W_0+\omega}$ for ${|\omega| < \delta}$ at ${x=\infty}$ for some ${\delta >0 }$ . Then, we prove that if the asymptotic state at left far field is given by ${{U_l}^{\pm}}$ , and if the perturbation of boundary of ${\mathcal{N}_0}$ and ${\delta}$ is sufficiently small, then there exists unique asymptotic state ${{U_r}^{\pm}}$ with a flat contact discontinuity ${\Gamma_{cd}^*}$ at right far field( ${x=\infty}$ ) and unique weak solution ${U}$ of the Euler system so that U consists of two subsonic flow with a contact discontinuity in between, and that U converges to ${{U_l}^{\pm}}$ and ${{U_r}^{\pm}}$ at ${x=-\infty}$ and ${x=\infty}$ respectively. For that purpose, we establish piecewise C ¹ estimate across a contact discontinuity of a weak solution to Euler system depending on the perturbation of ${\partial\mathcal{N}_0}$ and ${\delta}$ .     

Minimal subgroups and the structure of finite groups

A subgroup $H$ of a finite group $G$ is weakly-supplemented in $G$ if there exists a proper subgroup $K$ of $G$ such that $G=HK$ . In this paper we prove that a finite group $G$ is $p$ -nilpotent if every minimal subgroup of $P\bigcap G^{N}$ is weakly-supplemented in $G$ , and when $p=2$ either every cyclic subgroup of $P\bigcap G^{N}$ with order 4 is weakly-supplemented in $G$ or $P$ is quaternion-free, where $p$ is the smallest prime number dividing the order of $G$ , $P$ a sylow $p$ -subgroup of $G$ .