Third-Person Internalism: A Critical Examination of Externalism and a Foundation-Oriented Alternative

This paper starts with an examination of the major problems of foundation-oriented epistemology in Sect. 2. Then, in Sects. 3–4, it is argued that the externalistic re-definition of knowledge deprives this concept from useful applications to human’s epistemic practice. From the viewpoint of cultural evolution, the condition of justification is the most important ingredient of knowledge. An alternative foundation-oriented conception of knowledge called third-person internalism is developed in Sect. 2 and Sect. 5. It combines insights of externalism with the requirement of second-order justification. The application of third-person internalism to contextualistic positions leads to an important constraint on contextualism (Sect. 6). The final section (Sect. 7) sketches new prospects for a foundation-oriented epistemology which are based on epistemic optimality arguments.     

Many-Place Sequent Calculi for Finitely-Valued Logics

In this paper, we study multiplicative extensions of propositional many-place sequent calculi for finitely-valued logics arising from those introduced in Sect. 5 of Pynko (J Multiple-Valued Logic Soft Comput 10:339–362, 2004) through their translation by means of singularity determinants for logics and restriction of the original many-place sequent language. Our generalized approach, first of all, covers, on a uniform formal basis, both the one developed in Sect. 5 of Pynko (J Multiple-Valued Logic Soft Comput 10:339–362, 2004) for singular finitely-valued logics (when singularity determinants consist of a variable alone) and conventional Gentzen-style (i.e., two-place sequent) calculi suggested in Pynko (Bull Sect Logic 33(1):23–32, 2004) for finitely-valued logics with equality determinant. In addition, it provides a universal method of constructing Tait-style (i.e., one-place sequent) calculi for finitely-valued logics with singularity determinant (in particular, for Łukasiewicz finitely-valued logics) that fits the well-known Tait calculus (Lecture Notes in Mathematics, Springer, Berlin, 1968) for the classical logic. We properly extend main results of Pynko (J Multiple-Valued Logic Soft Comput 10:339–362, 2004) and explore calculi under consideration within the framework of Sect. 7 of Pynko (Arch Math Logic 45:267–305, 2006), generalizing the results obtained in Sect. 7.5 of Pynko (Arch Math Logic 45:267–305 2006) for two-place sequent calculi associated with finitely-valued logics with equality determinant according to Pynko (Bull Sect Logic 33(1):23–32, 2004). We also exemplify our universal elaboration by applying it to some denumerable families of well-known finitely-valued logics.     

Uniform Regularity in a Wedge and Regularity of Traces of CR Functions

We discuss in Sect. 1 the property of regularity at the boundary of separately holomorphic functions along families of discs and apply, in Sect. 2, to two situations. First, let W\mathcal{W} be a wedge of ℂ ⁿ with C ^ω , generic edge ℰ: a holomorphic function f on W\mathcal{W} has always a generalized (hyperfunction) boundary value bv(f) on ℰ, and this coincides with the collection of the boundary values along the discs which have C ^ω transversal intersection with ℰ. Thus Sect. 1 can be applied and yields the uniform continuity at ℰ of f when bv(f) is (separately) continuous. When W\mathcal{W} is only smooth, an additional property, the temperateness of f at ℰ, characterizes the existence of boundary value bv(f) as a distribution on ℰ. If bv(f) is continuous, this operation is consistent with taking limits along discs (Theorem 2.8). By Sect. 1, this yields again the uniform continuity at ℰ of tempered holomorphic functions with continuous bv. This is the theorem by Rosay (Trans. Am. Math. Soc. 297(1):63–72, 1986), in whose original proof the method of “slicing” by discs is not used.     

The twistor theory of the Hermitian Hurwitz pair (ℂ4(I 2,2),ℝ(I 2,3))

An analogue of the twistor theory is given for the Hermitian Hurwitz pair(ℂ⁴(I _2,2),ℝ(I _2,3)). In Sect. 2 a concept of Hurwitz twistors is introduced and a counterpart of the Penrose correspondence is obtained. It is proved that there exists a one-to-one correspondence between the twistors on the (1,3)-space and the (2,2)-space, which is called the duality theorem for Hurwitz twistors (Theorem 1). In Sect. 3. a concept of spinor equations is introduced for an Hermitian Hurwitz pair (abbreviated as HHP) and the duality theorem for solutions of the spinor equations is proved (Theorem 2). In Sect. 4 we give an elementary proof of the Penrose theory on the base of our Key Lemma. Then we can give the desired correspondence explicitly. In sect. 5 we consider the Penrose theory in the context of HHPs. At first we give a local version. It is proved that every solution of the spinor equation on the (2,2)-space can be represented as a ∂-harmonic one-form. By use of this result, we can get a direct relationship between the complex analysis and spinor theory on some open setM ⁺, which is called as “semi-global version” of the Penrose theory (Theorem 7). Moreover, we can get the original Penrose theory by use of the Penrose transformation (Theorem 5). Research of the first author partially supported by the State Committee for Scientific Research (KBN) grant PB 2 P03A 016 10 (Sections 1, 3 and 5 of the paper), and partially by the grant of the University of Łódź no. 505/485 (sections 2 and 4).     

A Renewal Theorem for Strongly Ergodic Markov Chains in Dimension <Emphasis Type="Italic">d</Emphasis> ≥ 3 and Centered Case

In dimension d ≥ 3, we present a general assumption under which the renewal theorem established by Spitzer (1964) for i.i.d. sequences of centered nonlattice r.v. holds true. Next we appeal to an operator-type procedure to investigate the Markov case. Such a spectral approach has been already developed by Babillot (Ann Inst Henri Poincaré, Sect B, Tome 24(4):507–569, 1988), but the weak perturbation theorem of Keller and Liverani (Ann Sc Norm Super Pisa CI Sci XXVIII(4):141–152, 1999) enables us to greatly weaken the moment conditions of Babillot (Ann Inst Henri Poincaré, Sect B, Tome 24(4):507–569, 1988). Our applications concern the v-geometrically ergodic Markov chains, the ρ-mixing Markov chains, and the iterative Lipschitz models, for which the renewal theorem of the i.i.d. case extends under the (almost) expected moment condition.     

Cryptosystems based on semi-distributive algebras

We propose a new cryptographic scheme of ElGamal type. The scheme is based on algebraic systems defined in the paper—semialgebras (Sect. 2). The main examples are semialgebras of polynomial mappings over a finite field K, and their factor-semialgebras. Given such a semialgebra R, one chooses an invertible element a ∈ R ^* of finite order r, and a random integer s. One chooses also a finite dimensional K-submodule V of R. The 4-tuple (R, V, a, b) where b = a ^s forms the public key for the cryptosystem, while r and s form the secret key. A plain text can be viewed as a sequence of elements of the field K. That sequence is divided into blocks of length dim(V) which, in turn, correspond to uniquely determined elements X _i of V. We propose three different methods (A, B, and C, see Definition 1.1) of encoding/decoding the sequence of X _i . The complexity of cracking the proposed cryptosystem is based on the Discrete Logarithm Problem for polynomial mappings (see Sect. 1.1). No methods of cracking the problem, except for the “brute force” (see Sect. 1.1) with Ω(r) time, are known so far.      

Complete asymptotic expansions associated with Epstein zeta-functions

Let Q(u,v)=|u+vz|² be a positive-definite quadratic form with a complex parameter z=x+iy in the upper-half plane. The Epstein zeta-function attached to Q is initially defined by for Re s>1, where the term with m=n=0 is to be omitted. We deduce complete asymptotic expansions of as y→+∞ (Theorem 1 in Sect. 2), and of its weighted mean value (with respect to y) in the form of a Laplace-Mellin transform of (Theorem 2 in Sect. 2). Prior to the proofs of these asymptotic expansions, the meromorphic continuation of over the whole s-plane is prepared by means of Mellin-Barnes integral transformations (Proposition 1 in Sect. 3). This procedure, differs slightly from other previously known methods of the analytic continuation, gives a new alternative proof of the Fourier expansion of (Proposition 2 in Sect. 3). The use of Mellin-Barnes type of integral formulae is crucial in all aspects of the proofs; several transformation properties of hypergeometric functions are especially applied with manipulation of these integrals. Research supported in part by Grant-in-Aid for Scientific Research (No. 13640041), the Ministry of Education, Culture, Sports, Science and Technology of Japan.     

König–Egerváry Graphs are Non-Edmonds

König–Egerváry graphs are those whose maximum matchings are equicardinal to their minimum-order coverings by vertices. Edmonds (J Res Nat Bur Standards Sect B 69B:125–130, 1965) characterized the perfect matching polytope of a graph G = (V, E) as the set of nonnegative vectors ${{\bf{x}}\in\mathbb R^E}K?nig–Egerváry graphs are those whose maximum matchings are equicardinal to their minimum-order coverings by vertices. Edmonds (J Res Nat Bur Standards Sect B 69B:125–130, 1965) characterized the perfect matching polytope of a graph G = (V, E) as the set of nonnegative vectors x ? \mathbb R^E{{\bf{x}}\in\mathbb R^E} satisfying two families of constraints: ‘vertex saturation’ and ‘blossom’. Graphs for which the latter constraints are implied by the former are termed non-Edmonds. This note presents two proofs—one combinatorial, one algorithmic—of its title’s assertion. Neither proof relies on the characterization of non-Edmonds graphs due to de Carvalho et al. (J Combin Theory Ser B 92:319–324, 2004).     

Defect Functions of Holomorphic Contractive Operator Functions and the Scattering Suboperator through the Internal Channels of a System. Part I

Let θ(ζ) be a Schur operator function, i.e., it is defined and holomorphic on the unit disk := C : 1 {\mathbb {D} := \{\zeta \in \mathbb {C} : \vert\zeta\vert < 1 \}} and its values are contractive operators acting from one Hilbert space into another one. In the first part of the paper the outer and *-outer Schur operator functions j(z){\varphi(\zeta)} and ψ(ζ) which describe respectively the deviations of the function θ(ζ) from inner and *-inner operator functions are studied. If j(z) 1 0{\varphi(\zeta)\neq 0} , then it means that in the scattering system for which θ(ζ) is the transfer function a portion of “information” comes inward the system and does not go outward, i.e., it is left in the internal channels of the system (Sect. 6). The function ψ(ζ) has the analogous property for the dual system. For this reason these functions are called the defect functions of the function θ(ζ). The explicit form of the defect functions j(z){\varphi(\zeta)} and ψ(ζ) is obtained and the analytic connection of these functions with the function θ(ζ) is described (Sects. 3, 5). The operator functions (l j(z)q(z)){\left(\begin{array}{l} \varphi(\zeta)\\ \theta(\zeta)\end{array}\right)} and (ψ(ζ), θ(ζ)) are Schur functions as well (Sect. 3). It is important that there exists the unique contractive measurable operator function χ(t), t ? ?\mathbb D{t\in\partial\mathbb {D}} , such that the operator function (l c(t)    j(t)y(t)    q(t) ){\left(\begin{array}{l} \chi(t)\quad \varphi(t)\\ \psi(t)\quad \theta(t) \end{array}\right)} , t ? ?\mathbb D,{t\in\partial\mathbb {D},} is also contractive (Part II, Sect. 12). The second part of the paper is devoted to studying the properties of the function χ(t). Specifically, it is shown that the function χ(t) is the scattering suboperator through the internal channels of the scattering system for which θ(ζ) is the transfer function (Part II, Sect. 12).     

Hyperbolic conservation laws with vanishing nonlinear diffusion and linear dispersion in heterogeneous media

We analyze family of solutions to multidimensional scalar conservation law, with flux depending on the time and space explicitly, regularized with vanishing diffusion and dispersion terms. Under a condition on the balance between diffusion and dispersion parameters, we prove that the family of solutions is precompact in L¹_loc{L^1_{\rm loc}}. Our proof is based on the methodology developed in Sazhenkov (Sibirsk Math Zh 47(2):431–454, 2006), which is in turn based on Panov’s extension (Panov and Yu in Mat Sb 185(2):87–106, 1994) of Tartar’s H-measures (Tartar in Proc R Soc Edinb Sect A 115(3–4):193–230, 1990), or Gerard’s micro-local defect measures (Gerard Commun Partial Differ Equ 16(11):1761–1794, 1991). This is new approach for the diffusion–dispersion limit problems. Previous results were restricted to scalar conservation laws with flux depending only on the state variable.     

Non-subelliptic estimates for the tangential Cauchy–Riemann system

We prove non-subelliptic estimates for the tangential Cauchy-Riemann system over a weakly “q-pseudoconvex” higher codimensional submanifold M of $\mathbb{C}^{n}We prove non-subelliptic estimates for the tangential Cauchy-Riemann system over a weakly “q-pseudoconvex” higher codimensional submanifold M of . Let us point out that our hypotheses do not suffice to guarantee subelliptic estimates, in general. Even more: hypoellipticity of the tangential C-R system is not in question (as shows the example by Kohn of (Trans AMS 181:273–292,1973) in case of a Levi-flat hypersurface). However our estimates suffice for existence of smooth solutions to the inhomogeneous C-R equations in certain degree. The main ingredients in our proofs are the weighted L ² estimates by H?rmander (Acta Math 113:89–152,1965) and Kohn (Trans AMS 181:273–292,1973) of Sect. 2 and the tangential -Neumann operator by Kohn of Sect 4; for this latter we also refer to the book (Adv Math AMS Int Press 19,2001). As for the notion of q pseudoconvexity we follow closely Zampieri (Compositio Math 121:155–162,2000). The main technical result, Theorem 2.1, is a version for “perturbed” q-pseudoconvex domains of a similar result by Ahn (Global boundary regularity of the -equation on q-pseudoconvex domains, Preprint, 2003) who generalizes in turn Chen-Shaw (Adv Math AMS Int Press 19, 2001).To Prof. Giovanni Zacher in his 80th birthday.     

Auxiliary functions in transcendental number theory

We discuss the role of auxiliary functions in the development of transcendental number theory. Initially, auxiliary functions were completely explicit (Sect. 1). The earliest transcendence proof is due to Liouville (Sect. 1.1) who produced the first explicit examples of transcendental numbers at a time where their existence was not yet known; in his proof, the auxiliary function is just a polynomial in one variable. Hermite’s proof of the transcendence of e (1873) is much more involved, the auxiliary function he builds (Sect. 1.2) is the first example of the Padé approximants (Sect. 1.3), which can be viewed as a far reaching generalization of continued fraction expansion (Brezinski in Lecture Notes in Math., vol. 888. Springer, Berlin, 1981; and Springer Series in Computational Mathematics, vol. 12. Springer, Berlin, 1991). Hypergeometric functions (Sect. 1.4) are among the best candidates for using Padé approximations techniques.     

Asymptotic expansions for a class of zeta-functions

In this paper, we shall reveal the hidden structure in recent results of Katsurada as the Meijer G-function hierarchy. In Sect. 1, we consider the holomorphic Eisenstein series and show that Katsurada’s two new expressions are variants of the classical Chowla–Selberg integral formula (Fourier expansion) with or without the beta-transform of Katsurada being incorporated. In Sect. 2, we treat the Taylor series expansion of the Lipschitz–Lerch transcendent in the perturbation variable. In the proofs, we make an extensive use of the beta-transform (used to be called the Mellin–Barnes formula).     

Translatable radii of an operator in the direction of another operator II

One of the couple of translatable radii of an operator in the direction of another operator introduced in earlier work [PAUL, K.: Translatable radii of an operator in the direction of another operator, Scientae Mathematicae 2 (1999), 119–122] is studied in details. A necessary and sufficient condition for a unit vector f to be a stationary vector of the generalized eigenvalue problem Tf = λAf is obtained. Finally a theorem of Williams ([WILLIAMS, J. P.: Finite operators, Proc. Amer. Math. Soc. 26 (1970), 129–136]) is generalized to obtain a translatable radius of an operator in the direction of another operator.     

Stratified structure of the Universe and Burgers' equation—a probabilistic approach

Summary The model of the potential turbulence described by the 3-dimensional Burgers' equation with random initial data was developped by Zeldovich and Shandarin, in order to explain the existing Large Scale Structure of the Universe. Most of the recent probabilistic investigations of large time asymptotics of the solution deal with the central limit type results (the Gaussian scenario), under suitable moment assumptions on the initial velocity field. These results and some open questions are discussed in Sect. 2, where we concentrate on the Gaussian model and the shot-noise model. In Sect. 3 we construct a probabilistic model of strong initial fluctuations (a zero-range shot-noise field with high amplitudes) which reveals an intermittent large time behaviour, with the velocity determined by the position of the largest initial fluctuation (discounted by the heat kernelg(t,x·)) in a neighborhood ofx. The asymptoties of such local maximum ast can be analyzed with the help of the theory of records (Sect. 4). Finally, in Sect. 5 we introduce a global definition of a point process oft-local maxima, and show the weak convergence of the suitably rescaled process to a non-trivial limit ast.     

Characteristic properties of large subgroups in primary abelian groups

SupposeG is an arbitrary additively written primary abelian group with a fixed large subgroupL. It is shown thatG is (a) summable; (b) σ-summable; (c) a Σ-group; (d) p^ω+1-projecrive only when so isL. These claims extend results of such a kind obtained by Benabdallah, Eisenstadt, Irwin and Poluianov,Acta Math. Acad. Sci. Hungaricae (1970) and Khan,Proc. Indian Acad. Sci. Sect. A (1978).     

On regularity of a boundary point for higher-order parabolic equations: towards Petrovskii-type criterion by blow-up approach

The classic problem of regularity of boundary points for higher-order partial differential equations (PDEs) is concerned. For second-order elliptic and parabolic equations, this study was completed by Wiener’s (J. Math. Phys. Mass. Inst. Tech. 3:127–146, 1924) and Petrovskii’s (Math. Ann. 109:424–444, 1934) criteria, and was extended to more general equations including quasilinear ones. Since the 1960–1970s, the main success was achieved for 2mth-order elliptic PDEs; e.g., by Kondrat’ev and Maz’ya. However, the higher-order parabolic ones, with infinitely oscillatory kernels, were not studied in such details. As a basic model, explaining typical difficulties of regularity issues, the 1D bi-harmonic equation in a domain shrinking to the origin (0, 0) is concentrated upon:

Variational conditions with smooth constraints: structure and analysis

 This is an expository paper about the analysis of variational conditions over sets defined in finite-dimensional spaces by fairly smooth functions satisfying a constraint qualification. The primary focus is on results that can provide quantitative and computable sensitivity information for particular instances of the problems under study, and our objective is to give a personal view of the state of current knowledge in this area and of gaps in that knowledge that require future work. The writing style is informal, in keeping with the objective of focusing the reader's attention on the basic concepts and the relationships between them, rather than on details of the particular results themselves. Received: December 1, 2002 / Accepted: April 25, 2003 Published online: May 28, 2003 Key words. variational condition – variational inequality – complementarity – sensitivity – stability – nondegeneracy Mathematics Subject Classification (2000): Primary: 90C31. Secondary: 47J20, 49J40, 49J53, 90C33