Universal rigidity of bar frameworks via the geometry of spectrahedra

A bar framework (G, p) in dimension r is a graph G whose nodes are points \(p^1,\ldots ,p^n\) in \(\mathbb {R}^r\) and whose edges are line segments between pairs of these points. Two frameworks (G, p) and (G, q) are equivalent if each edge of (G, p) has the same (Euclidean) length as the corresponding edge of (G, q). A pair of non-adjacent vertices i and j of (G, p) is universally linked if \(||p^i-p^j||\) = \(||q^i-q^j||\) in every framework (G, q) that is equivalent to (G, p). Framework (G, p) is universally rigid iff every pair of non-adjacent vertices of (G, p) is universally linked. In this paper, we present a unified treatment of the universal rigidity problem based on the geometry of spectrahedra. A spectrahedron is the intersection of the positive semidefinite cone with an affine space. This treatment makes it possible to tie together some known, yet scattered, results and to derive new ones. Among the new results presented in this paper are: (1) The first sufficient condition for a given pair of non-adjacent vertices of (G, p) to be universally linked. (2) A new, weaker, sufficient condition for a framework (G, p) to be universally rigid thus strengthening the existing known condition. An interpretation of this new condition in terms of the Strong Arnold Property is also presented.     

Semiclassical resonances associated with a periodic orbit

We consider resonances for a h-pseudo-differential operator H(x, hD _x; h) induced by a periodic orbit of hyperbolic type. We generalize the framework of Gérard and Sjöstrand, in the sense that we allow hyperbolic and elliptic eigenvalues of the Poincarémap, and look for so-called semi-excited resonances with imaginary part of magnitude ?h log h, or h ^δ, with 0 < δ < 1.     

Koszul property of a class of graded algebras with nonpure resolutions

Given any integers a, b, c, and d with a > 1, c ≥ 0, b ≥ a + c, and d ≥ b + c, the notion of (a, b, c, d)-Koszul algebra is introduced, which is another class of standard graded algebras with “nonpure” resolutions, and includes many Artin-Schelter regular algebras of low global dimension as specific examples. Some basic properties of (a, b, c, d)-Koszul algebras/modules are given, and several criteria for a standard graded algebra to be (a, b, c, d)-Koszul are provided.     

On collineation groups of finite projective spaces containing a Singer cycle

By a result of Kantor, any subgroup of GL(n, q) containing a Singer cycle normalizes a field extension subgroup. This result has as a consequence a projective analogue, and this paper gives the details of this deduction, showing that any subgroup of PΓL(n, q) containing a projective Singer cycle normalizes the image of a field extension subgroup GL(n/s, q^s) under the canonical homomorphism GL(n, q) → PGL(n, q), for some divisor s of n, and so is contained in the image of ΓL(n/s, q^s) under the canonical homomorphism ΓL(n, q) → PΓL(n, q). The actions of field extension subgroups on V (n, q) are also investigated. In particular, we prove that any field extension subgroup GL(n/s, q^s) of GL(n, q) has a unique orbit on s-dimensional subspaces of V (n, q) of length coprime to q. This orbit is a Desarguesian s-partition of V (n, q).     

Regular variation of a random length sequence of random variables and application to risk assessment

When assessing risks on a finite-time horizon, the problem can often be reduced to the study of a random sequence C(N) = (C ₁,…,C _N ) of random length N, where C(N) comes from the product of a matrix A(N) of random size N × N and a random sequence X(N) of random length N. Our aim is to build a regular variation framework for such random sequences of random length, to study their spectral properties and, subsequently, to develop risk measures. In several applications, many risk indicators can be expressed from the extremal behavior of ∥C(N)∥, for some norm ∥?∥. We propose a generalization of Breiman’s Lemma that gives way to a tail estimate of ∥C(N)∥ and provides risk indicators such as the ruin probability and the tail index for Shot Noise Processes on a finite-time horizon. Lastly, we apply our main result to a model used in dietary risk assessment and in non-life insurance mathematics to illustrate the applicability of our method.     

Heat Kernels for Isotropic-Like Markov Generators on Ultrametric Spaces: a Survey

Let (X, d) be a locally compact separable ultrametric space. Let D be the set of all locally constant functions having compact support. Given a measure m and a symmetric function J(x, y) we consider the linear operator L^Jf(x) = ∫(f(x) ? f(y)) J(x, y)dm(y) defined on the set D. When J(x, y) is isotropic and satisfies certain conditions, the operator (?L^J, D) acts in L²(X,m), is essentially self-adjoint and extends as a self-adjoint Markov generator, its Markov semigroup admits a continuous heat kernel p^J (t, x, y). When J(x, y) is not isotropic but uniformly in x, y is comparable to isotropic function J(x, y) as above the operator (?L^J, D) extends in L²(X,m) as a self-adjointMarkov generator, its Markov semigroup admits a continuous heat kernel p^J(t, x, y), and the function p^J(t, x, y) is uniformly comparable in t, x, y to the function p^J(t, x, y), the heat kernel related to the operator (?L^J,D).     

Arnold diffusion for a complete family of perturbations

In this work we illustrate the Arnold diffusion in a concrete example — the a priori unstable Hamiltonian system of 2 + 1/2 degrees of freedom H(p, q, I, φ, s) = p ²/2+ cos q ? 1 + I ²/2 + h(q, φ, s; ε) — proving that for any small periodic perturbation of the form h(q, φ, s; ε) = ε cos q (a ₀₀ + a ₁₀ cosφ + a ₀₁ cos s) (a ₁₀ a ₀₁ ≠ 0) there is global instability for the action. For the proof we apply a geometrical mechanism based on the so-called scattering map. This work has the following structure: In the first stage, for a more restricted case (I* ~ π/2μ, μ = a ₁₀/a ₀₁), we use only one scattering map, with a special property: the existence of simple paths of diffusion called highways. Later, in the general case we combine a scattering map with the inner map (inner dynamics) to prove the more general result (the existence of instability for any μ). The bifurcations of the scattering map are also studied as a function of μ. Finally, we give an estimate for the time of diffusion, and we show that this time is primarily the time spent under the scattering map.     

On the Distribution of Zero Sets of Holomorphic Functions

Let M be a subharmonic function with Riesz measure ν _M in a domain D in the n-dimensional complex Euclidean space ? _n , and let f be a nonzero function that is holomorphic in D, vanishes on a set Z ? D, and satisfies |f| ? expM on D. Then restrictions on the growth of ν _M near the boundary of D imply certain restrictions on the dimensions or the area/volume of Z. We give a quantitative study of this phenomenon in the subharmonic framework.     

Designs associated with maximum independent sets of a graph

A (v, β _o , μ)-design over regular graph G = (V, E) of degree d is an ordered pair D = (V, B), where |V| = v and B is the set of maximum independent sets of G called blocks such that if i, j ∈ V, i ≠ j and if i and j are not adjacent in G then there are exactly μ blocks containing i and j. In this paper, we study (v, β _o , μ)-designs over the graphs K _n × K _n , T(n)-triangular graphs, L ₂(n)-square lattice graphs, Petersen graph, Shrikhande graph, Clebsch graph and the Schläfli graph and non-existence of (v, β _o , μ)-designs over the three Chang graphs T ₁(8), T ₂(8) and T ₃(8).     

On a classical theorem on the diameter and minimum degree of a graph

In this work, we obtain good upper bounds for the diameter of any graph in terms of its minimum degree and its order, improving a classical theorem due to Erd¨os, Pach, Pollack and Tuza.We use these bounds in order to study hyperbolic graphs(in the Gromov sense). To compute the hyperbolicity constant is an almost intractable problem, thus it is natural to try to bound it in terms of some parameters of the graph. Let H(n, δ_0) be the set of graphs G with n vertices and minimum degree δ_0, and J(n, Δ) be the set of graphs G with n vertices and maximum degree Δ. We study the four following extremal problems on graphs: a(n, δ_0) = min{δ(G) | G ∈ H(n, δ_0)}, b(n, δ_0) = max{δ(G) |G ∈ H(n, δ_0)}, α(n, Δ) = min{δ(G) | G ∈ J(n, Δ)} and β(n, Δ) = max{δ(G) | G ∈ J(n, Δ)}. In particular, we obtain bounds for b(n, δ_0) and we compute the precise value of a(n, δ_0), α(n, Δ) andβ(n, Δ) for all values of n, δ_0 and Δ, respectively.     

On the number of edges in a uniform hypergraph with a range of permitted intersections

The paper studies the quantity p(n, k, t ₁, t ₂) equal to the maximum number of edges in a k-uniform hypergraph with the property that the size of the intersection of any two edges lies in the interval [t ₁, t ₂]. Previously known upper and lower bounds are given. New bounds for p(n, k, t ₁, t ₂) are obtained, and the relationship between these bounds and known estimates is studied. For some parameter values, the exact values of p(n, k, t ₁, t ₂) are explicitly calculated.     

Finite Groups with a Given Set of Character Degrees

Let G be a finite group with {1, q, r, qm, q ² m, rqm} as the character degree set, where r and q are distinct primes and m>1 is an integer not divisible by q or r. We show that G is solvable and the derived length of G equals 3.     

The Relaxed Edge-Coloring Game and <Emphasis Type="Italic">k</Emphasis>-Degenerate Graphs

The (r, d)-relaxed edge-coloring game is a two-player game using r colors played on the edge set of a graph G. We consider this game on forests and more generally, on k-degenerate graphs. If F is a forest with Δ(F)=Δ, then the first player, Alice, has a winning strategy for this game with r=Δ?j and d≥2j+2 for 0≤j≤Δ?1. This both improves and generalizes the result for trees in Dunn, C. (Discret. Math. 307, 1767–1775, 2007). More broadly, we generalize the main result in Dunn, C. (Discret. Math. 307, 1767–1775, 2007) by showing that if G is k-degenerate with Δ(G)=Δ and j∈[Δ+k?1], then there exists a function h(k,j) such that Alice has a winning strategy for this game with r=Δ+k?j and d≥h(k,j).     

A modified large-scale structure-preserving doubling algorithm for a large-scale Riccati equation from transport theory

A large scale nonsymmetric algebraic Riccati equation XCX ? XE ? AX + B = 0 arising in transport theory is considered, where the n × n coefficient matrices B,C are symmetric and low-ranked and A, E are rank one updates of nonsingular diagonal matrices. By introducing a balancing strategy and setting appropriate initial matrices carefully, we can simplify the large-scale structure-preserving doubling algorithm (SDA_ls) for this special equation. We give modified large-scale structure-preserving doubling algorithm, which can reduce the flop count of original SDA_ls by half. Numerical experiments illustrate the effectiveness of our method.     

Kostka functions associated to complex reflection groups and a conjecture of Finkelberg-Ionov

Kostka functions K_(λ,μ)~±(t), indexed by r-partitions λ and μ of n, are a generalization of Kostka polynomials K_(λ,μ)(t) indexed by partitions λ,μ of n. It is known that Kostka polynomials have an interpretation in terms of Lusztig's partition function. Finkelberg and Ionov(2016) defined alternate functions K_(λ,μ)(t) by using an analogue of Lusztig's partition function, and showed that K_(λ,μ)(t) ∈Z≥0[t] for generic μ by making use of a coherent realization. They conjectured that K_(λ,μ)(t) coincide with K_(λ,μ)~-(t). In this paper, we show that their conjecture holds. We also discuss the multi-variable version, namely, r-variable Kostka functions K_(λ,μ)~±(t_1,…,t_r).     

Coherence relative to a weak torsion class

Let R be a ring. A subclass T of left R-modules is called a weak torsion class if it is closed under homomorphic images and extensions. Let T be a weak torsion class of left R-modules and n a positive integer. Then a left R-module M is called T-finitely generated if there exists a finitely generated submodule N such that M/N ∈ T; a left R-module A is called (T,n)-presented if there exists an exact sequence of left R-modules

$$0 \to {K_{n - 1}} \to {F_{n - 1}} \to \cdots \to {F_1} \to {F_0} \to M \to 0$$

such that F₀,..., F_n?1 are finitely generated free and K_n?1 is T-finitely generated; a left R-module M is called (T,n)-injective, if Ext ⁿ _R (A,M) = 0 for each (T, n+1)-presented left R-module A; a right R-module M is called (T,n)-flat, if Tor ^R _n (M,A) = 0 for each (T, n+1)-presented left R-module A. A ring R is called (T,n)-coherent, if every (T, n+1)-presented module is (n + 1)-presented. Some characterizations and properties of these modules and rings are given.

     

Bound states of a two-boson system on a two-dimensional lattice

We consider a Hamiltonian of a two-boson system on a two-dimensional lattice Z². The Schrödinger operator H(k₁, k₂) of the system for k₁ = k₂ = π, where k = (k₁, k₂) is the total quasimomentum, has an infinite number of eigenvalues. In the case of a special potential, one eigenvalue is simple, another one is double, and the other eigenvalues have multiplicity three. We prove that the double eigenvalue of H(π,π) splits into two nondegenerate eigenvalues of H(π, π ? 2β) for small β > 0 and the eigenvalues of multiplicity three similarly split into three different nondegenerate eigenvalues. We obtain asymptotic formulas with the accuracy of β² and also an explicit form of the eigenfunctions of H(π, π ?2β) for these eigenvalues.     

Theory of Semi-Instantiation in Abstract Argumentation

We study instantiated abstract argumentation frames of the form (S, R, I), where (S, R) is an abstract argumentation frame and where the arguments x of S are instantiated by I(x) as well formed formulas of a well known logic, for example as Boolean formulas or as predicate logic formulas or as modal logic formulas. We use the method of conceptual analysis to derive the properties of our proposed system. We seek to define the notion of complete extensions for such systems and provide algorithms for finding such extensions. We further develop a theory of instantiation in the abstract, using the framework of Boolean attack formations and of conjunctive and disjunctive attacks. We discuss applications and compare critically with the existing related literature.     

A group of continuous self-maps on a topological groupoid

The group of bisections of groupoids plays an important role in the study of Lie groupoids. In this paper another construction is introduced. Indeed, for a topological groupoid G, the set of all continuous self-maps f on G such that (x, f(x)) is a composable pair for every \(x\in G\), is denoted by \(S_G\). We show that \(S_G\) by a natural binary operation is a monoid. \(S_G(\alpha )\), the group of units in \(S_G\) precisely consists of those \(f\in S_G\) such that the map \(x\mapsto xf(x)\) is a bijection on G. Similar to the group of bisections, \(S_G(\alpha )\) acts on G from the right and on the space of continuous self-maps on G from the left. It is proved that \(S_G(\alpha )\) with the compact- open topology inherited from C(G, G) is a left topological group. For a compact Hausdorff groupoid G it is proved that the group of bisections of \(G^2\) is isomorphic to the group \(S_G(\alpha )\) and the group of transitive bisections of G, \(Bis_T(G)\), is embedded in \(S_G(\alpha )\), where \(G^2\) is the groupoid of all composable pairs.     

The Corona problem on a complete ultrametric algebraically closed field

Let IK be a complete ultrametric algebraically closed field and let A be the Banach IK-algebra of bounded analytic functions in the ”open” unit disk D of IK provided with the Gauss norm. Let Mult(A, ‖. ‖) be the set of continuous multiplicative semi-norms of A provided with the topology of simple convergence, let Mult _m (A, ‖. ‖) be the subset of the ? ∈ Mult(A, ‖. ‖) whose kernel is amaximal ideal and let Mult ₁(A, ‖. ‖) be the subset of the ? ∈ Mult(A, ‖. ‖) whose kernel is a maximal ideal of the form (x ? a)A with a ∈ D. By analogy with the Archimedean context, one usually calls ultrametric Corona problem the question whether Mult ₁(A, ‖. ‖) is dense in Mult _m (A, ‖. ‖). In a previous paper, it was proved that when IK is spherically complete, the answer is yes. Here we generalize this result to any algebraically closed complete ultrametric field, which particularly applies to ? _p . On the other hand, we also show that the continuous multiplicative seminorms whose kernel are neither a maximal ideal nor the zero ideal, found by Jesus Araujo, also lie in the closure of Mult ₁(A, ‖. ‖), which suggest that Mult ₁(A, ‖. ‖)might be dense in Mult(A, ‖. ‖).