Search

Frieze patterns have been introduced by Coxeter [Acta Arith. 18 (1971), pp. 297–310] in the 1970's and have recently attracted renewed interest due to their close connection with Fomin-Zelevinsky's cluster algebras. Frieze patterns can be interpreted as assignments of values to the diagonals of a triangulated polygon satisfying certain conditions for crossing diagonals (Ptolemy relations). Weak friezes, as introduced by Çanakçı and Jørgensen [Adv. in Appl. Math. 131 (2021), Paper No. 102253], are generalizing this concept by allowing to glue dissected polygons so that the Ptolemy relations only have to be satisfied for crossings involving one of the gluing diagonals. To any frieze pattern one can associate a symmetric matrix using a triangular fundamental domain of the frieze pattern in the upper and lower half of the matrix and putting zeroes on the diagonal. Broline, Crowe and Isaacs [Geometriae Dedicata 3 (1974), pp. 171–176] have found a formula for the determinants of these matrices and their work has later been generalized in various directions by other authors. These frieze pattern determinants are the main focus of our paper. As our main result we show that this determinant behaves well with respect to gluing weak friezes: the determinant is the product of the determinants for the pieces glued, up to a scalar factor coming from the gluing diagonal. Then we give several applications of this result, showing that formulas from the literature, obtained by Broline-Crowe-Isaacs, Baur-Marsh [J. Combin. Theory Ser. A 119 (2012), pp. 1110–1122], Bessenrodt-Holm-Jørgensen [J. Combin. Theory Ser. A 123 (2014), pp. 30–42] and Maldonado [ Frieze matrices and infinite frieze patterns with coefficients , Preprint, arXiv: 2207.04120 , 2022] can all be obtained as consequences of our result. [ABSTRACT FROM AUTHOR]

In this paper, we study parameter deformations of matrix valued orthogonal polynomials. These deformations are built on the use of certain matrix valued operators which are symmetric with respect to the matrix valued inner product defined by the orthogonality weight. We show that the recurrence coefficients associated with these operators satisfy generalizations of the non-Abelian lattice equations. We provide a Lax pair formulation for these equations, and an example of deformed Hermite-type matrix valued polynomials is discussed in detail. [ABSTRACT FROM AUTHOR]

Let S be the polynomial ring on the space of non-square generic matrices or the space of odd-sized skew-symmetric matrices, and let I be the determinantal ideal of maximal minors or Pf the ideal of sub-maximal Pfaffians, respectively. Using desingularizations and representation theory of the general linear group we expand upon work of Raicu–Weyman–Witt [Adv. Math. 250 (2014), pp. 596–610] to determine the S-module structures of Ext^j_S(S/I^t, S) and Ext^j_S(S/Pf^t, S), from which we get the degrees of generators of these Ext modules. As a consequence, via graded local duality we answer a question of Wenliang Zhang [J. Pure Appl. Algebra 225 (2021), Paper No. 106789] on the socle degrees of local cohomology modules of the form H^j_\mathfrak {m}(S/I^t). [ABSTRACT FROM AUTHOR]

In the structural theory of cluster algebras, a crucial role is played by a family of integer vectors, called c-vectors, which parametrize the coefficients. It has recently been shown that each c-vector with respect to an acyclic initial seed is a real root of the corresponding root system. In this paper, we obtain an interpretation of this result in terms of symmetric matrices. We show that for skew-symmetric cluster algebras, the c-vectors associated with any seed defines a quasi-Cartan companion for the corresponding exchange matrix (i.e. they form a companion basis), and we establish some basic combinatorial properties. In particular, we show that these vectors define an admissible cut of edges in the associated quivers. [ABSTRACT FROM AUTHOR]

A characterization of convex functions in RN states that an upper semicontinuous function u is convex if and only if u(Ax) is subharmonic (with respect to the usual Laplace operator) for every symmetric positive definite matrix A. The aim of this paper is to prove that an analogue of this result holds for free Carnot groups G when considering convexity in the viscosity sense. In the subelliptic context of Carnot groups, the linear maps x … Ax of the Euclidean case must be replaced by suitable group isomorphisms x … TA(x), whose differential preserves the first layer of the stratification of Lie(G). [ABSTRACT FROM AUTHOR]

Let Sn be the set of all n-by-n symmetric real matrices, and let Cn be the copositive cone, that is, the set of all matrices a ∈ Sn that fulfill the condition utop a u ≥ 0 for all n-vectors u with nonnegative entries. We prove that a linear mapping φ : Sn → Sn satisfies φ (Cn)= Cn if and only if φ (x)=mtop xm for a fixed monomial matrix m with nonnegative entries. [ABSTRACT FROM AUTHOR]

The Tamari lattice Tn was originally defined on bracketings of a set of n+1 objects, with a cover relation based on the associativity rule in one direction. Although in several related lattices the number of maximal chains is known, quoting Knuth, "The enumeration of such paths in Tamari lattices remains mysterious". The lengths of maximal chains vary over a great range. In this paper, we focus on the chains with maximum length in these lattices. We establish a bijection between the maximum length chains in the Tamari lattice and the set of standard shifted tableaux of staircase shape. We thus derive an explicit formula for the number of maximum length chains, using the Thrall formula for the number of shifted tableaux. We describe the relationship between chains of maximum length in the Tamari lattice and certain maximal chains in weak Bruhat order on the symmetric group, using standard Young tableaux. Additionally, recently Bergeron and Préville-Ratelle introduced a generalized Tamari lattice. Some of the results mentioned above carry over to their generalized Tamari lattice. [ABSTRACT FROM AUTHOR]

Let M_n be drawn uniformly from all \pm 1 symmetric n \times n matrices. We show that the probability that M_n is singular is at most \exp (-c(n\log n)^{1/2}), which represents a natural barrier in recent approaches to this problem. In addition to improving on the best-known previous bound of Campos, Mattos, Morris and Morrison of \exp (-c n^{1/2}) on the singularity probability, our method is different and considerably simpler: we prove a "rough" inverse Littlewood-Offord theorem by a simple combinatorial iteration. [ABSTRACT FROM AUTHOR]

Let XR be the zero locus in RPn of one or two independently and Kostlan distributed random real quadratic forms (this is equivalent to the corresponding symmetric matrices being in the Gaussian Orthogonal Ensemble). Denoting by b(XR) the sum of the Betti numbers of XR, we prove that (1)...The methods we use combine random matrix theory, integral geometry and spectral sequences: for one quadric hypersurface it is simply a corollary of Wigner's semicircle law; for the intersection of two quadrics it is related to the (intrinsic) volume of the set of singular symmetric matrices of norm one. [ABSTRACT FROM AUTHOR]

A criterion for a contraction T on a Hilbert space to be complex symmetric is given in terms of the operator-valued characteristic function ΘT of T in 2007 (see Nicolas Chevrot, Emmanuel Fricain, and Dan Timotin [Proc. Amer. Math. Soc. 135 (2007), pp. 2877–2886]). To further classify unitary equivalent complex symmetric contractions, we notice a simple condition of when ΘT1 and ΘT2 coincide for two complex symmetric contractions T1 and T2. As an application, surprisingly we solve the problem for any defect index n, when the defect indexes of contractions are 2, this problem was left open by Nicolas Chevrot, Emmanuel Fricain, and Dan Timotin [Proc. Amer. Math. Soc. 135 (2007), pp. 2877–2886]. Furthermore, a construction of 3 × 3 symmetric inner matrices is proposed, which extends some results on 2 × 2 inner matrices (see Stephan Ramon Garcia [J. Operator Theory 54 (2005), pp. 239–250]) and 2 × 2 symmetric inner matrices (see Nicolas Chevrot, Emmanuel Fricain, and Dan Timotin [Proc. Amer. Math. Soc. 135 (2007), pp. 2877–2886]). [ABSTRACT FROM AUTHOR]

Let D be the unit disc and z0 ϵ D. We determine the value range {f(z0) | f ϵ R≥}, where R≥ is the set of holomorphic functions f : D → D with f(0) = 0 and f'(0) ≥ 0 that have only real coefficients in their power series expansion around 0, and the smaller set {f(z0) | f ϵ R≥, f is typically real}. Furthermore, we describe a third value range {f(z0) | f ϵ I}, where I consists of all univalent self-mappings of the upper half-plane H with hydrodynamical normalization which are symmetric with respect to the imaginary axis. [ABSTRACT FROM AUTHOR]

We adapt the inverse iteration method for symmetric matrices to some nonlinear PDE eigenvalue problems. In particular, for p ∈ (1, ∞) and a given domain Ω ⊂ ℝn, we analyze a scheme that allows us to approximate the smallest value the ratio ∫Ω |Dψ|pdx/ ∫ Ω|ψ|pdx can assume for functions ψ that vanish on ∂Ω. The scheme in question also provides a natural way to approximate minimizing ψ. Our analysis also extends in the limit as p → ∞ and thereby fashions a new approximation method for ground states of the infinity Laplacian. [ABSTRACT FROM AUTHOR]