Approximation of Functions of Few Variables in?High?Dimensions

Let f be a continuous function defined on Ω:=[0,1] ^N which depends on only ℓ coordinate variables, f(x₁,?,x_N)=g(x_i1,?,x_il)f(x_{1},\ldots,x_{N})=g(x_{i_{1}},\ldots,x_{i_{\ell}}). We assume that we are given m and are allowed to ask for the values of f at m points in Ω. If g is in Lip1 and the coordinates i ₁,…,i _ℓ are known to us, then by asking for the values of f at m=L ^ℓ uniformly spaced points, we could recover f to the accuracy |g|_Lip1 L ⁻¹ in the norm of C(Ω). This paper studies whether we can obtain similar results when the coordinates i ₁,…,i _ℓ are not known to us. A prototypical result of this paper is that by asking for C(ℓ)L ^ℓ (log ₂ N) adaptively chosen point values of f, we can recover f in the uniform norm to accuracy |g|_Lip1 L ⁻¹ when g∈Lip1. Similar results are proven for more general smoothness conditions on g. Results are also proven under the assumption that f can be approximated to some tolerance ε (which is not known) by functions of ℓ variables.     

What Norm of Assertion?

I argue that the debates over which norm constitutes assertion can be abandoned by challenging the three main motivations for a constitutive norm. The first motivation is the alleged analogy between language and games. The second motivation is the intuition that some assertions are worthy of criticism. The third is the discursive responsibilities incurred by asserting. I demonstrate that none of these offer good reasons to believe in a constitutive norm of assertion, as such a norm is understood in the literature. Others who have made similar arguments conclude that assertion does not exist at all—that there is no such thing as assertion. I disagree: we do not have to relinquish the category of assertion just because it is not normatively constituted. There are alternative ways to understand and individuate assertion that do not rely on a constitutive norm.     

Preservation of Absolutely Continuous Spectrum of?Periodic Jacobi Operators Under Perturbations of?Square-Summable Variation

We study self-adjoint bounded Jacobi operators of the form:

Various disconnectivities of spaces and projectabilities of ?-groups

Arch denotes the category of archimedean ?-groups and ?-homomorphisms. Tych denotes the category of Tychonoff spaces with continuous maps, and α denotes an infinite cardinal or ∞. This work introduces the concept of an αcc-disconnected space and demonstrates that the class of αcc-disconnected spaces forms a covering class in Tych. On the algebraic side, we introduce the concept of an αcc-projectable ?-group and demonstrate that the class of αcc-projectable ?-groups forms a hull class in Arch. In addition, we characterize the αcc-projectable objects in W—the category of Arch-objects with designated weak unit and ?-homomorphisms that preserve the weak unit—and construct the αcc-hull for G in W. Lastly, we apply our results to negatively answer the question of whether every hull class (resp., covering class) is epireflective (resp., monocoreflective) in the category of W-objects with complete ?-homomorphisms (resp., the category of compact Hausdorff spaces with skeletal maps).     

A generalization of a basic inequality of Telyakovski?

Two classes of numerical sequences are defined by means certain properties of the differences of terms of positive sequences, and their relationship to some newly defined classes and the well-known Sidon-Telyakovski? class is analyzed. It is also verified that if a sequence belongs to the newly defined wider class, then with this sequence in place of the sequence {1/k} an essential inequality established by Telyakovski? can be generalized notably. This new result and the previous generalizations of the original Telyakovski???s theorem are incomparable.     

Continuous on rays solutions of an equation of the Go?a?b-Schinzel type

Let X be a real linear space. We characterize continuous on rays solutions f,g:X→R of the equation f(x+g(x)y)=f(x)f(y). Our result refers to papers of J. Chudziak (2006) [14] and J. Brzd?k (2003) [11].     

Moments of Ramanujan?s generalized elliptic integrals and extensions of Catalan?s constant

We undertake a thorough investigation of the moments of Ramanujan?s alternative elliptic integrals and of related hypergeometric functions. Along the way we are able to give some surprising closed forms for Catalan-related constants and various new hypergeometric identities.     

An Equivalent of Algebraic ?-Categories

In this paper, based on ?-categories, some properties of(continuous) I-cocomplete ?-categories are studied. Then, we introduce the concepts of bicomplete ?-category and approximable bimodule, discuss their properties and we also show any I-cocomplete ?-category is a bicomplete ?-category. Finally, it is proved that the category of algebraic ?-categories is equivalent to the category of bicomplete ?-categories.     

On the completeness of systems of eigenfunctions and associated functions of differential operators of orders 2 ? �� and 1 ? ��

In the space of vector-functions, we consider a boundary-value problem for differential operators of fractional orders (2 ? ??) and (1 ? ??) and prove the completeness of the system of eigenfunctions and associated functions of this problem in the space $L_1 \left( {\left[ {0,1} \right],\,\mathbb{C}^p } \right)$ .     

Analytic Continuations of Fourier and Stieltjes Transforms and Generalized Moments of?Probability?Measures

We consider analytic continuations of Fourier transforms and Stieltjes transforms. This enables us to define what we call complex moments for some class of probability measures which do not have moments in the usual sense. There are two ways to generalize moments accordingly to Fourier and Stieltjes transforms; however these two turn out to coincide. As applications, we give short proofs of the convergence of probability measures to Cauchy distributions with respect to tensor, free, Boolean and monotone convolutions.     

The End of Moral Realism?

The author considers how constructivism, presently known to us essentially as a theory for generating rules of social cooperation, embodies a certain conception of justification that in turn may be thought of as a general theory. It is argued that moral realism and projectivism are by turns platitudinous and unsatisfactory as conceptions of justification; by contrast the general conception of justification in constructivism makes sense of reason giving and coherent rivalry. The author argues that once the right picture of justification is in place, the picture constructivism illustrates or embodies, the problem of moral ontology disappears.

On solutions of a common generalization of the Go?a?b-Schinzel equation and of the addition formulae

Under some additional assumptions we determine solutions of the equation

f(x+M(f(x))y)=f(x)○f(y),     

Independence of ? in Lafforgue's theorem

Let X be a smooth curve over a finite field of characteristic p, let ?≠p be a prime number, and let be an irreducible lisse -sheaf on X whose determinant is of finite order. By a theorem of L. Lafforgue, for each prime number ?′≠p, there exists an irreducible lisse -sheaf on X which is compatible with , in the sense that at every closed point x of X, the characteristic polynomials of Frobenius at x for and are equal. We prove an “independence of ?” assertion on the fields of definition of these irreducible ?′-adic sheaves : namely, that there exists a number field F such that for any prime number ?′≠p, the -sheaf above is defined over the completion of F at one of its ?′-adic places.     

Asymptotic Form of Hermite-Pad? Polynomials

Permanent address: Department of Physics, Saint Louis University, St Louis, Missouri, U.S.A. A method is given which locates the area on which the zerosof Hermite-Pad? polynomials of large order lie. The method alsogives the density of these zeros along the ares, and so theasymptotic form of the polynomials for large order. Severalexamples which confirm and illustrate the method are given.These results are necessary for proving the convergence of schemesfor extracting information from power series based on Hermite-Pad?approximants (the so-called "quadratic" and "differential equation"or "integral" Pad? approximants).     

Ehrhart h ?-Vectors of Hypersimplices

We consider the Ehrhart h ^?-vector for the hypersimplex. It is well-known that the sum of the $h_{i}^{*}$ is the normalized volume which equals the Eulerian numbers. The main result is a proof of a conjecture by R.?Stanley which gives an interpretation of the $h^{*}_{i}$ coefficients in terms of descents and exceedances. Our proof is geometric using a careful book-keeping of a shelling of a unimodular triangulation. We generalize this result to other closely related polytopes.     

Hochschild cohomology of ? n -Galois coverings of an algebra

We consider the ? _n -Galois covering ?? _n of the algebra A introduced by F. Xu [Adv. Math., 2008, 219: 1872?C1893]. We calculate the dimensions of all Hochschild cohomology groups of ?? _n and give the ring structure of the Hochschild cohomology ring modulo nilpotence. As a conclusion, we provide a class of counterexamples to Snashall-Solberg??s conjecture.     

Some of Melvin Henriksen?s contributions to spaces of ideals

The three of us have written this note to discuss Mel Henriksen?s joint paper with us, Joincompact spaces, continuous lattices andC^?-algebras. In this paper we learned that the space of closed primal ideals of a C^?-algebra is a continuous lattice, so it is joincompact when equipped with the lower and Scott topologies. In addition to its mathematics, we discuss how Mel brought us into this work, and his and its continuing influence on us.     

Commutators of Riesz Transforms Related to?Schr?dinger Operators

In this work we obtain boundedness on L ^p , for 1<p<??, of commutators T _b f=bTf?T(bf) where T is any of the Riesz transforms or their conjugates associated to the Schr?dinger operator ???+V with V satisfying an appropriate reverse H?lder inequality. The class where b belongs is larger than the usual BMO. We also obtain a substitute result for p=??, under a slightly stronger condition on?b.     

Existence and regularity of positive solutions of elliptic equations of Schr?dinger type

We prove the existence of positive solutions with optimal local regularity of the homogeneous equation of Schr?dinger type $$ - {\rm{div}}(A\nabla u) - \sigma u = 0{\rm{ in }}\Omega $$ for an arbitrary open ?? ? ? ⁿ under only a form-boundedness assumption on ?? ?? D??(??) and ellipticity assumption on A ?? L ^??(??) ^n×n . We demonstrate that there is a two-way correspondence between form boundedness and existence of positive solutions of this equation as well as weak solutions of the equation with quadratic nonlinearity in the gradient $$ - {\rm{div}}(A\nabla u) = (A\nabla v) \cdot \nabla v + \sigma {\rm{ in }}\Omega $$ As a consequence, we obtain necessary and sufficient conditions for both formboundedness (with a sharp upper form bound) and positivity of the quadratic form of the Schr?dinger type operator H = ?div(A?·)-?? with arbitrary distributional potential ?? ?? D??(??), and give examples clarifying the relationship between these two properties.     

On a core concept of Arhangel'ski?

Arhangel'ski? [A.V. Arhangel'ski?, Locally compact spaces of countable core and Alexandroff compactification, Topology Appl. 154 (2007) 625-634] has introduced a weakening of σ-compactness: having a countable core, for locally compact spaces, and asked when it is equivalent to σ-compactness. We settle several problems related to that paper.