Abstract In this paper, we develop new Hermite-Hadamard-Mercer type inequalities on coordinates via post-quantum calculus, also known as (p, q) - calculus. By introducing novel (p1, p2, q1, q2)-differentiable and (p1, p2, q1, q2)-integrable functions, we generalize classical results and extend previous inequalities under the setting of coordinate convexity. Several new identities are derived, which naturally reduce to known results when specific parameters are chosen. Numerical examples and visualizations are also provided to illustrate the utility of our results.
For a cardinal lambda<lambda_{omega_1} we give a ccc forcing notion P which forces that for some Borel subset B of the Cantor space (1) there a sequence (eta_alpha:alpha<lambda) of distinct elements such that |(eta_alpha+B) cap (eta_beta+B)|>5 for all alpha,beta<lambda, but (2) there is no perfect set of such eta's. The construction closely follows the one from Shelah math.LO/9802134
Using the method of decisive creatures (math.LO/0601083) we show the consistency of "there is no increasing omega_2 --chain of Borel sets and non(N)=non(M)= omega_2=2^omega". Hence, consistently, there are no monotone hulls for the ideal M cap N . This answers Balcerzak and Filipczak. Next we use FS iteration with partial memory to show that there may be monotone Borel hulls for the ideals M, N even if they are not generated by towers.
We study varieties with a term-definable poset structure, "po-groupoids". It is known that connected posets have the "strict refinement property" (SRP). In [arXiv:0808.1860v1 [math.LO]] it is proved that semidegenerate varieties with the SRP have definable factor congruences and if the similarity type is finite, directly indecomposables are axiomatizable by a set of first-order sentences. We obtain such a set for semidegenerate varieties of connected po-groupoids and show its quantifier complexity is bounded in general.
We try to redo, improve and continue the non-structure parts in some works on a.e.c., which uses weak diamond, in lambda^+ and lambda^{++} getting better and more results and do what is necessary for the book on a.e.c. Comparing with math.LO/9805146 we make the context closer to the examples, hence hopefully improve transparency, though losing some generality. Toward this we work also on the positive theory, i.e. structure side of "low frameworks" like almost good lambda-frames.
In a recent paper [6], this author has extended the method of the kernel function [1] to the boundary value problems of the generalized axially symmetric potentialsThis method can also be applied to a more general class of singular differential equations, namelyor, equivalently,We shall derive in the sequel explicit formulas for the Dirichlet problems of (1.1) in the first quadrant of the x-y plane in terms of sufficiently smooth boundary data, and obtain an error-bound for their approximate solutions. We shall also indicate how the Neumann problem can be solved.
In this paper we present some approximation theorems for the eigenvalue problem of a compact linear operator defined on a Banach space. In particular we examine: criteria for the existence and convergence of approximate eigenvectors and generalized eigenvectors; relations between the dimensions of the eigenmanifolds and generalized eigenmanifolds of the operator and those of the approximate operators.
We investigate the existence of strong colorings on successors of singular cardinals. This work continues Section 2 of [Sh:413] (math.LO/9809199), but now our emphasis is on finding colorings of pairs of ordinals, rather than colorings of finite sets of ordinals.
We give a survey with some explanations but no proofs of the new notion of b-minimality by the author and F. Loeser [b-minimality, J. Math. Log., 7 no. 2 (2007) 195--227, math.LO/0610183]. We compare this notion with other notions like o-minimality, C-minimality, p-minimality, and so on.
Given a class of finite models we would like to expand each model (allowing new elements but the old universe is a separate sort), making the expressive power of LFP (least fix point logic) and PFP (inductive logic) similar while not changing the expressive power of FO (first order logic). This continues in math.LO/9411235.
The main result of this paper is a partial answer to [math.LO/9909115, Problem 5.5]: a finite iteration of Universal Meager forcing notions adds generic filters for many forcing notions determined by universality parameters. We also give some results concerning cardinal characteristics of the sigma-ideals determined by those universality parameters.
We further investigate the class of models of a strongly dependent (first order complete) theory T, continuing math.LO/0406440. If |A|+|T|<= mu, I subseteq C, |I| >=beth_{|T|^+}(mu) then some J subseteq I of cardinality mu^+ is an indiscernible sequence over A .
Solecki proved that the group of automorphisms of a countable structure cannot be an uncountable free abelian group. See more in Just, Shelah and Thomas math.LO/0003120 where as a by product we can say something on on uncountable structures. We prove here the following Theorem: If A is a countable model, then Aut(M) cannot be a free uncountable group.
We introduce more properties of forcing notions which imply that their lambda-support iterations are lambda-proper, where lambda is an inaccessible cardinal. This paper is a direct continuation of section A.2 of math.LO/0210205. As an application of our iteration result we show that it is consistent that dominating numbers associated with two normal filters on lambda are distinct.
Vinay Deolalikar, Joel David Hamkins, Ralf-Dieter Schindler
Extending results of Schindler [math.LO/0106087] and Hamkins and Welch [math.LO/0212046], we establish in the context of infinite time Turing machines that P is properly contained in NP intersect coNP. Furthermore, NP intersect coNP is exactly the class of hyperarithmetic sets. For the more general classes, we establish that P+ = (NP+ intersect coNP+) = (NP intersect coNP), though P++ is properly contained in NP++ intersect coNP++. Within any contiguous block of infinite clockable ordinals, we show that P_alpha is not equal to NP_alpha intersect coNP_alpha, but if beta begins a gap in the clockable ordinals, then P_beta = NP_beta intersect coNP_beta. Finally, we establish that P^f is not equal to NP^f intersect coNP^f for most functions f from the reals to the ordinals, although we provide examples where P^f = NP^f intersect coNP^f and P^f is not equal to NP^f.