An Example of two Cardinals that are Equivalent in the n-Order Logic and not Equivalent in the (n + 1)-Order Logic

It is proved that the property of two models to be equivalent in the nth order logic is definable in the (n + 1)th order logic. Basing on this fact, there is given an (nonconstructive) “example” of two n-order equivalent cardinal numbers that are not (n + 1)-order equivalent.     

Local sentences and Mahlo cardinals

Local sentences were introduced by Ressayre in [6] who proved certain remarkable stretching theorems establishing the equivalence between the existence of finite models for these sentences and the existence of some infinite well ordered models. Two of these stretching theorems were only proved under certain large cardinal axioms but the question of their exact (consistency) strength was left open in [4]. Here we solve this problem, using a combinatorial result of J. H. Schmerl [7]. In fact, we show that the stretching principles are equivalent to the existence of n ‐Mahlo cardinals for appropriate integers n. This is done by proving first that for every integer n, there is a local sentence φ_n having well ordered models of order type τ, for every infinite ordinal τ > ω which is not an n ‐Mahlo cardinal. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On extensions of supercompactness

We show that, in terms of both implication and consistency strength, an extendible with a larger strong cardinal is stronger than an enhanced supercompact, which is itself stronger than a hypercompact, which is itself weaker than an extendible. All of these are easily seen to be stronger than a supercompact. We also study Cⁿ‐supercompactness.     

具有正负周期系数的差分方程的振动性与线性化振动性

本文得到具有正负周期系数的差分方程振动的一个充分必要条件,并利用此充要条件获得了一个线性化振动性结果。     

Simple formulas defining complicated sets

We consider the question whether large cardinal axioms imply that certain complicated sets cannot be defined by simple formulas. More precisely, we ask whether the existence of larger large cardinals is compatible with the existence of a well-ordering of the real numbers that is definable by a Σ₁-formula that uses a single ordinal as a parameter. This note presents results by Ralf Schindler, Philipp Schlicht and the author showing that the existence of a well-ordering of the reals that is definable by a Σ₁-formula with parameter ω₁ is compatible with the existence of a Woodin cardinal and incompatible with the existence of a Woodin cardinal with a measurable cardinal above it. Moreover, a similar result holds for Σ₁-formulas using certain large cardinals as a parameter. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Output Feedback Guaranteed Cost Control for Uncertain Discrete-Time Systems Using Linear Matrix Inequalities

This paper considers the output feedback guaranteed cost controller design problem for uncertain discrete-time systems. The uncertainty is assumed to be of linear fractional form, unstructured, and is allowed to be time-varying. It is proved that the existence of a guaranteed cost controller is equivalent to the feasibility of a certain linear matrix inequality (LMI), and that a full-order output feedback controller can be constructed in terms of the feasible solution to the LMI. Furthermore, a convex optimization problem is introduced for the selection of a suitable controller minimizing a specified cost bound.     

On the Samarskii-Andreev transmission conditions in the theory of elastic beams

It is proved that the transmission conditions for elastic beams in the case of a nonideal joint are limiting in the construction of asymptotics for the transmission problem for two thin elastic bodies if the boundary between the bodies is filled by slightly extendible material.     

Existence of an A-optimal model for a regression experiment

The existence of certain A-optimal models for a regression experiment is equivalent to the solvability of certain matrix equations. It is proved that for any m × m matrix V (over the real field), there exists an orthogonal matrix Q such that QVQ′ has equal diagonal elements. This result is used to solve the equations mentioned above. The connection of this result to Hadamard matrices is discussed.     

DISCRETE UNIFORM STRUCTURES AND QUASITOPOI

It is known that no non-trivial subcategory of the category of topological spares is a quasitopos in the sense of Penon. The purpose of this note is to establish the existence of a proper class of sub-categories of the category of uniform spaces which are quasitopoi. These subcategories are generated by certain proximally discrete uniform spaces which correspond to each infinite regular cardinal.     

Implications between strong large cardinal axioms

The rank-into-rank and stronger large cardinal axioms assert the existence of certain elementary embeddings. By the preservation of the large cardinal properties of the embeddings under certain operations, strong implications between various of these axioms are derived.     

Embedding (r, 1)-designs in finite projective planes

It is known that any non-trivial (r,1)-design on υ varieties (υ ? (r? 1)² ? 1) is extendible; this fact implies the existence of a projective plane of order r ? 1. In this paper it is shown that any non-trivial (r, 1)-design on (r ? 1)² ? α varieties, where r and α are appropriately bounded, is extendible; hence this fact implies the existence of a projective plane of order r ? 1. We also show that, for υ ? (r ? 1)² ? 2, any non-trivial (r, 1)-design on υ varieties is extendible.     

The theorem of the means for cardinal and ordinal numbers

The theorem that the arithmetic mean is greater than or equal to the geometric mean is investigated for cardinal and ordinal numbers. It is shown that whereas the theorem of the means can be proved for n pairwise comparable cardinal numbers without the axiom of choice, the inequality a² + b² ≥ 2ab is equivalent to the axiom of choice. For ordinal numbers, the inequality α² + β² ≥ 2αβ is established and the conditions for equality are derived; stronger inequalities are obtained for finite and infinite sequences of ordinals under suitable monotonicity hypotheses. MSC: 03E10, 04A10, 03E25, 04A25.     

There exist subdirectly irreducible commutative basic algebras of an arbitrary infinite cardinality which are not MV-algebras

It was recently proved by P. Wojciechowski that for any infinite cardinal there exists a linearly ordered MV-algebra of this cardinality. Since basic algebras are a (non-associative) generalization of MV-algebras, there rises a natural question if this is true also for basic algebras which are not MV-algebras. Using the construction by P. Wojciechowski and the modified construction by the first author, we can set up certain defectors which enable us to prove the result of the title.     

Infinite horizon disturbance attenuation for discrete-time systems. A Popov-Yakubovich approach

It is proved that for the discrete-time linear systems with time-varying coefficients the existence of a controller which simultaneously stabilizes and provides prescribed disturbance attenuation for the resultant closed-loop system, implies the existence of global solutions to several Kalman-Szegö-Popov-Yakubovich systems. It is also proved that this fact is equivalent to the existence of the positive semidefinite stabilizing solutions to corresponding game-theoretic Riccati equations. The family of all controllers with the above mentioned properties is constructed in terms of the solutions to the cited Kalman-Szegö-Popov-Yakubovich systems. The main tool is the generalized Popov-Yakubovich theory which is essentially developed in an operator-theoretic framework.     

Small embedding characterizations for large cardinals

We show that many large cardinal notions can be characterized in terms of the existence of certain elementary embeddings between transitive set-sized structures, that map their critical point to the large cardinal in question. As an application, we use such embeddings to provide new proofs of results of Christoph Weiß on the consistency strength of certain generalized tree properties. These new proofs eliminate problems contained in the original proofs provided by Weiß.     

Minimization of Gerstewitz functionals extending a scalarization by Pascoletti and Serafini

Scalarization in vector optimization is often closely connected to the minimization of Gerstewitz functionals. In this paper, the minimizer sets of Gerstewitz functionals are investigated. Conditions are given under which such a set is nonempty and compact. Interdependencies between solutions of problems with different parameters or with different feasible point sets are shown. Consequences for the parameter control in scalarization methods are proved. It is pointed out that the minimization of Gerstewitz functionals is equivalent to an optimization problem which generalizes the scalarization by Pascoletti and Serafini. The results contain statements about minimizers of certain Minkowski functionals and norms. Some existence results for solutions of vector optimization problems are derived.     

Logics that define their own semantics

The capability of logical systems to express their own satisfaction relation is a key issue in mathematical logic. Our notion of self definability is based on encodings of pairs of the type (structure, formula) into single structures wherein the two components can be clearly distinguished. Hence, the ambiguity between structures and formulas, forming the basis for many classical results, is avoided. We restrict ourselves to countable, regular, logics over finite vocabularies. Our main theorem states that self definability, in this framework, is equivalent to the existence of complete problems under quantifier free reductions. Whereas this holds true for arbitrary structures, we focus on examples from Finite Model Theory. Here, the theorem sheds a new light on nesting hierarchies for certain generalized quantifiers. They can be interpreted as failure of self definability in the according extensions of first order logic. As a further application we study the possibility of the existence of recursive logics for PTIME. We restate a result of Dawar concluding from recursive logics to complete problems. We show that for the model checking Turing machines associated with a recursive logic, it makes no difference whether or not they may use built in clocks. Received: 7 February 1997     

Banach spaces that admit support sets

It is shown that the existence of a closed convex set all of whose points are properly supported in a Banach space is equivalent to the existence of a certain type of uncountable ordered one-sided biorthogonal system. Under the continuum hypothesis, we deduce that this notion is weaker than the existence of an uncountable biorthogonal system.