Characterizing Convexity of a Function by Its Fréchet and Limiting Second-Order Subdifferentials

The Fréchet and limiting second-order subdifferentials of a proper lower semicontinuous convex function \(\varphi: \mathbb R^n\rightarrow\bar{\mathbb R}\) have a property called the positive semi-definiteness (PSD)—in analogy with the notion of positive semi-definiteness of symmetric real matrices. In general, the PSD is insufficient for ensuring the convexity of an arbitrary lower semicontinuous function φ. However, if φ is a C ^1,1 function then the PSD property of one of the second-order subdifferentials is a complete characterization of the convexity of φ. The same assertion is valid for C ¹ functions of one variable. The limiting second-order subdifferential can recognize the convexity/nonconvexity of piecewise linear functions and of separable piecewise C ² functions, while its Fréchet counterpart cannot.     

A Midpoint Method for Generalized Equations Under Mild Differentiability Condition

The aim of this study is the approximation of a solution x ^? of the generalized equation 0∈f(x)+F(x) in Banach spaces, where f is a single function whose second order Fréchet derivative ?² f verifies an Hölder condition, and F stands for a set-valued map with closed graph. Using a fixed point theorem and proceeding by induction under the pseudo-Lipschitz property of F, we obtain a sequence defined by a midpoint formula whose convergence to x ^? is superquadratic. Taking a weaker condition, we present the result obtained when ?² f satisfies a center-Hölder conditioning.     

Partition of a unity on infinite-dimensional manifold of the Lipschiz class Lip<Superscript><Emphasis Type="Italic">k</Emphasis></Superscript>

In the present paper we prove a criterion of Lip ^k -paracompactness for infinitedimensional manifold M modeled in nonnormable topological vector Fréchet space F. We establish that a manifold is Lip ^k -paracompact if and only if the model space F is paracompact and Lip ^k -normal. We prove a sufficient condition for existence of Lip ^k -partition of a unity on a manifold of class Lip ^k .     

Bivariate Binomial Moments and Bonferroni-Type Inequalities

We obtain bivariate forms of Gumbel’s, Fréchet’s and Chung’s linear inequalities for P(S ≥ u, T ≥ v) in terms of the bivariate binomial moments {S _{i, j} }, 1 ≤ i ≤ k,1 ≤ j ≤ l of the joint distribution of (S, T). At u = v = 1, the Gumbel and Fréchet bounds improve monotonically with non-decreasing (k, l). The method of proof uses combinatorial identities, and reveals a multiplicative structure before taking expectation over sample points.     

Schauder Decompositions and the Grothendieck and Dunford-Pettis Properties in Köthe Echelon Spaces of Infinite Order

It is shown that every echelon space λ_∞(A), with A an arbitrary Köthe matrix, is a Grothendieck space with the Dunford-Pettis property. Since λ_∞(A) is Montel if and only if it coincides with λ₀(A), this identifies an extensive class of non-normable, non-Montel Fréchet spaces having these two properties. Even though the canonical unit vectors in λ_∞(A) fail to form an unconditional basis whenever λ_∞(A) ≠ λ₀(A), it is shown, nevertheless, that in this case λ_∞(A) still admits unconditional Schauder decompositions (provided it satisfies the density condition). This is in complete contrast to the Banach space setting, where Schauder decompositions never exist. Consequences for spectral measures are also given.     

Approximating the Fréchet Distance for Realistic Curves in Near Linear Time

We present a simple and practical (1+ε)-approximation algorithm for the Fréchet distance between two polygonal curves in ? ^d . To analyze this algorithm we introduce a new realistic family of curves, c-packed curves, that is closed under simplification. We believe the notion of c-packed curves to be of independent interest. We show that our algorithm has near linear running time for c-packed polygonal curves, and similar results for other input models, such as low-density polygonal curves.     

Cyclic cohomology of certain nuclear Fréchet algebras and DF algebras

We give explicit formulae for the continuous Hochschild and cyclic homology and cohomology of certain \(\hat \otimes\)-algebras. We use well-developed homological techniques together with some niceties of the theory of locally convex spaces to generalize the results known in the case of Banach algebras and their inverse limits to wider classes of topological algebras. To this end we show that, for a continuous morphism ?: x → y of complexes of complete nuclear DF-spaces, the isomorphism of cohomology groups H ⁿ(?): H ⁿ (x) → H ⁿ (y) is automatically topological. The continuous cyclic-type homology and cohomology are described up to topological isomorphism for the following classes of biprojective \(\hat \otimes\)-algebras: the tensor algebra E \(\hat \otimes\) F generated by the duality (E,F,<·,·>) for nuclear Fréchet spaces E and F or for nuclear DF-spaces E and F; nuclear biprojective Köthe algebras λ(P) which are Fréchet spaces or DF-spaces; the algebra of distributions ε^*(G) on a compact Lie group G.     

Grothendieck spaces with the Dunford–Pettis property

Banach spaces which are Grothendieck spaces with the Dunford–Pettis property (briefly, GDP) are classical. A systematic treatment of GDP-Fréchet spaces occurs in Bonet and Ricker (Positivity 11:77–93, 2007). This investigation is continued here for locally convex Hausdorff spaces. The product and (most) inductive limits of GDP-space are again GDP-spaces. Also, every complete injective space is a GDP-space. For \({p\in \{0\}\cup[1,\infty)}\) it is shown that the classical co-echelon spaces k _p (V) and \({K_p(\overline{V})}\) are GDP-spaces if and only if they are Montel. On the other hand, \({K_\infty(\overline{V})}\) is always a GDP-space and k _∞(V) is a GDP-space whenever its (Fréchet) predual, i.e., the Köthe echelon space λ ₁(A), is distinguished.     

Inversion and subspaces of a finite field

Consider two F _q -subspaces A and B of a finite field, of the same size, and let A ^?1 denote the set of inverses of the nonzero elements of A. The author proved that A ^?1 can only be contained in A if either A is a subfield, or A is the set of trace zero elements in a quadratic extension of a field. Csajbók refined this to the following quantitative statement: if A ^?1 ? B, then the bound |A ^?1∩B| ≤ 2|B|/q ? 2 holds. He also gave examples showing that his bound is sharp for |B| ≤ q ³. Our main result is a proof of the stronger bound |A ^?1 ∩ B| ≤ |B|/q · (1 + O _d (q ^?1/2)), for |B| = q ^d with d > 3. We also classify all examples with |B| ≤ q ³ which attain equality or near-equality in Csajbók’s bound.     

Duality between Fréchet differentiability and strong convexity

This paper revisits the duality between differentiability and strict or strong convexity under the Legendre–Fenchel transform \({f\mapsto f^*}\). Functions f defined on a Banach space X are considered. For a lower semicontinuous but not necessarily convex function f we relate essential Fréchet differentiability of the conjugate function f* to essential strong convexity of f.     

The kernel of the second order Cauchy difference on semigroups

Let S be a semigroup, H a 2-torsion free, abelian group and \(C^2f\) the second order Cauchy difference of a function \(f:S \rightarrow H\). Assuming that H is uniquely 2-divisible or S is generated by its squares we prove that the solutions f of \(C^2f = 0\) are the functions of the form \(f(x) = j(x) + B(x,x)\), where j is a solution of the symmetrized additive Cauchy equation and B is bi-additive. Under certain conditions we prove that the terms j and B are continuous, if f is. We relate the solutions f of \(C^2f = 0\) to Fréchet’s functional equation and to polynomials of degree less than or equal to 2.     

Pricing <Emphasis Type="Italic">k</Emphasis><Superscript>th</Superscript> realization derivatives and collateralized debt obligation with multivariate Fréchet copula

Copula method has been widely applied to model the correlation among underlying assets in financial market. In this paper, we propose to use the multivariate Fréchet copula family presented in J. P. Yang et al. [Insurance Math. Econom., 2009, 45: 139–147] to price multivariate financial instruments whose payoffs depend on the k ^th realization of the underlying assets and collateralized debt obligation (CDO). The advantage of the multivariate Fréchet copula is discussed. Empirical study shows that such copula family gives a better fitting to CDO’s market price than Gaussian copula for some derivatives.     

Acute perturbation of Drazin inverse and oblique projectors

For an n×n complex matrix A with ind(A) = r; let A^D and A^π = I–AA^D be respectively the Drazin inverse and the eigenprojection corresponding to the eigenvalue 0 of A: For an n×n complex singular matrix B with ind(B) = s, it is said to be a stable perturbation of A, if I–(B^π–A^π)² is nonsingular, equivalently, if the matrix B satisfies the condition \(\mathcal{R}(B^s)\cap\mathcal{N}(A^r)=\left\{0\right\}\) and \(\mathcal{N}(B^s)\cap\mathcal{R}(A^r)=\left\{0\right\}\), introduced by Castro-González, Robles, and Vélez-Cerrada. In this paper, we call B an acute perturbation of A with respect to the Drazin inverse if the spectral radius ρ(B^π–A^π) < 1: We present a perturbation analysis and give suffcient and necessary conditions for a perturbation of a square matrix being acute with respect to the matrix Drazin inverse. Also, we generalize our perturbation analysis to oblique projectors. In our analysis, the spectral radius, instead of the usual spectral norm, is used. Our results include the previous results on the Drazin inverse and the group inverse as special cases and are consistent with the previous work on the spectral projections and the Moore-Penrose inverse.     

Twisted homology for the mirabolic nilradical

The notion of derivatives for smooth representations of GL(n, ? _p ) was defined in [BZ77]. In the archimedean case, an analog of the highest derivative was defined for irreducible unitary representations in [Sah89] and called the “adduced” representation. In [AGS] derivatives of all orders were defined for smooth admissible Fréchet representations (of moderate growth).A key ingredient of this definition is the functor of twisted coinvariants with respect to the nilradical of the mirabolic subgroup. In this paper we prove exactness of this functor and compute it on a certain class of representations. This implies exactness of the highest derivative functor, and allows to compute highest derivatives of all monomial representations.In [AGS] these results are applied to finish the computation of adduced representations for all irreducible unitary representations and to prove uniqueness of degenerate Whittaker models for unitary representations.     

A note on generating <Emphasis Type="Italic">P</Emphasis>-matrices

We prove that for any \({A,B\in\mathbb{R}^{n\times n}}\) such that each matrix S satisfying min(A, B) ≤ S ≤ max(A, B) is nonsingular, all four matrices A ^?1 B, AB ^?1, B ^?1 A and BA ^?1 are P-matrices. A practical method for generating P-matrices is drawn from this result.     

Elliptic function of level 4

The article is devoted to the theory of elliptic functions of level n. An elliptic function of level n determines a Hirzebruch genus called an elliptic genus of level n. Elliptic functions of level n are also of interest because they are solutions of the Hirzebruch functional equations. The elliptic function of level 2 is the Jacobi elliptic sine function, which determines the famous Ochanine–Witten genus. It is the exponential of the universal formal group of the form F(u, v) = (u² ? v²)/(uB(v) ? vB(u)), B(0) = 1. The elliptic function of level 3 is the exponential of the universal formal group of the form F(u, v) = (u²A(v) ? v²A(u))/(uA(v)² ? vA(u)²), A(0) = 1, A″(0) = 0. In the present study we show that the elliptic function of level 4 is the exponential of the universal formal group of the form F(u, v) = (u²A(v) ? v²A(u))/(uB(v) ? vB(u)), where A(0) = B(0) = 1 and for B′(0) = A″(0) = 0, A′(0) = A₁, and B″(0) = 2B₂ the following relation holds: (2B(u) + 3A₁u)² = 4A(u)³ ? (3A₁² ? 8B₂)u²A(u)². To prove this result, we express the elliptic function of level 4 in terms of the Weierstrass elliptic functions.     

The abstract Dirichlet problem for continuous vector-valued functions

Let X be a simplex with the set \({\text {ext}}\, X\) of extreme points being Lindelöf. We show that a continuous bounded mapping defined on \({\text {ext}}\, X\) with values in a Fréchet space F can be extended to a mapping that is a pointwise limit of a bounded sequence of affine continuous mappings from X to F.     

Asymptotic behavior for log-determinants of several non-Hermitian random matrices

We study the asymptotic behavior for log-determinants of two unitary but non-Hermitian random matrices: the spherical ensembles A ^-1 B; where A and B are independent complex Ginibre ensembles and the truncation of circular unitary ensembles. The corresponding Berry-Esseen bounds and Cramér type moderate deviations are established. Our method is based on the estimates of corresponding cumulants. Numerical simulations are also presented to illustrate the theoretical results.     

An improvement to double-step Newton method and its multi-step version for solving system of nonlinear equations and its applications

In this work, we have improved the order of the double-step Newton method from four to five using the same number of evaluation of two functions and two first order Fréchet derivatives for each iteration. The multi-step version requires one more function evaluation for each step. The multi-step version converges with order 3r+5, r≥1. Numerical experiments are done comparing the new methods with some existing methods. Our methods are also tested on Chandrasekhar’s problem and the 2-D Bratu problem to illustrate the applications.