Measurable cardinals and good Σ1(κ)\Sigma_1(\kappa)-wellorderings

We study the influence of the existence of large cardinals on the existence of wellorderings of power sets of infinite cardinals $\kappa$ with the property that the collection of all initial segments of the wellordering is definable by a $\Sigma_1$-formula with parameter $\kappa$. A short argument shows that the existence of a measurable cardinal $\delta$ implies that such wellorderings do not exist at $\delta$-inaccessible cardinals of cofinality not equal to $\delta$ and their successors. In contrast, our main result shows that these wellorderings exist at all other uncountable cardinals in the minimal model containing a measurable cardinal. In addition, we show that measurability is the smallest large cardinal property that interferes with the existence of such wellorderings at uncountable cardinals and we generalize the above result to the minimal model containing two measurable cardinals.Comment: 14 page

Algebraic genericity of strict-order integrability

We provide sharp conditions on a measure µ defined on a measurable space X guaranteeing that the family of functions in the Lebesgue space Lp (µ, X) (p ≥ 1) which are not integrable with order q for any q > p (or any q < p) contains, except for zero, large subspaces of Lp (µ, X). This improves recent results due to Aron, García, Muñoz, Palmberg, Pérez, Puglisi and Seoane. It is also shown that many nonintegrable functions of order q can be obtained even on any nonempty open subset of X, assuming that X is a topological space and µ is a Borel measure on X satisfying appropriate properties.Plan Andaluz de Investigación (Junta de Andalucía)Ministerio de Ciencia e InnovaciónMinisterio de Ciencia y Tecnología (MCYT). Españ

Коаксиальная тепловая труба для охлаждения отражателя лазера

Описаны результаты разработки и исследования коаксиальной тепловой трубы для охлаждения отражателя твердотельного лазера. Показано, что система охлаждения, функционирующая по испарительно-конденсационному принципу, позволяет обеспечить равномерность температуры охлаждаемой поверхности при низком термическом сопротивлении 0,03 К/Вт.Наведено результати розробки та дослідження коаксіальної теплової труби для охолодження відбивача твердотільного лазера. Система охолодження, яка функціонує за випаровувально-конденсаційним принципом, дозволяє забезпечити рівномірність температури охолоджуваної поверхні при низькому термічному опорі.The paper presents the development and research results for coaxial heat designed for cooling reflector of a solid-state laser. A coaxial cylindrical heat pipe, designed to cool the laser reflector, provides that the temperature of the heat-removing surface does not exceed 120°C at any orientation in the gravitational field, if the heat is removed by forced convection of air with the temperature of 60°C in a pulsed mode of heat flow supply of 300 W. Thermal resistance of the developed heat pipe is 0,03 K/W, the specific thermal resistance — 1,1•10⁻³ m²•К/W. The developed cooling system based on the evaporation-condensation principle, allows ensuring temperature uniformity of the cooling surface at low thermal resistance

Local club condensation and L-likeness

AbstractWe present a forcing to obtain a localized version of Local Club Condensation, a generalized Condensation principle introduced by Sy Friedman and the first author in [3] and [5]. This forcing will have properties nicer than the forcings to obtain this localized version that could be derived from the forcings presented in either [3] or [5]. We also strongly simplify the related proofs provided in [3] and [5]. Moreover our forcing will be capable of introducing this localized principle at κ while simultaneously performing collapses to make κ become the successor of any given smaller regular cardinal. This will be particularly useful when κ has large cardinal properties in the ground model. We will apply this to measure how much L-likeness is implied by Local Club Condensation and related principles. We show that Local Club Condensation at κ+ is consistent with ¬☐κ whenever κ is regular and uncountable, generalizing and improving a result of the third author in [14], and that if κ ≥ ω2 is regular, CC(κ+) - Chang’s Conjecture at κ+ - is consistent with Local Club Condensation at κ+, both under suitable large cardinal consistency assumptions.</jats:p

Condensation and large cardinals

Wir definieren lokale Clubmengenkondensation (Local Club Condensation), ein Prinzip, welches Eigenschaften von Gödels Kondensationsprinzip isoliert und verallgemeinert. Wir zeigen, dass wir über einem beliebigen Modell der Mengenlehre durch die Erzwingungsmethode zu einem Modell der Mengenlehre gelangen können, welches lokale Clubmengenkondensation erfüllt und zugleich verschiedene grosse Kardinalzahlen erhalten werden können; insbesondere zeigen wir, dass lokale Clubmengenkondensation mit der Existenz einer omega-superstarken Kardinalzahl konsistent ist. Wir gehen ähnlich für Acceptability vor, ein weiteres Prinzip welches Aspekte von Gödels Kondensationsprinzip isoliert und verallgemeinert. Dies setzt das Outer Model Program (zu deutsch Programm der äusseren Modelle) von Sy Friedman fort. Wir führen auch eine mögliche Zukunftsanwendung in Bezug auf das Proper Forcing Axiom PFA oben beschriebener Ergebnisse am Ende der Arbeit auf.We define Local Club Condensation, a principle which isolates and generalizes properties of Gödel's Condensation principle. We show that we can force over any model of set theory to obtain a model which satisfies this principle while at the same time preserving various very large cardinals; in particular we show that Local Club Condensation is consistent with the existence of an omega-superstrong cardinal. We proceed similarly for Acceptability, another principle isolating and generalizing aspects of Gödel's Condensation principle. This continues the outer model program of Sy Friedman. We also hint at a possible future application of the above-described results at the end of this thesis regarding the consistency strength of PFA

Approximation Property of Functions and the Absolute Nörlund Summability of Fourier Series

Article信州大学工学部紀要 43: 1-10 (1977)departmental bulletin pape

Downward transference of mice and universality of local core models

If M is a proper class inner model of ZFC and omega_2^M=omega_2, then every sound mouse projecting to omega and not past 0-pistol belongs to M. In fact, under the assumption that 0-pistol does not belong to M, K^M \| omega_2 is universal for all countable mice in V. Similarly, if M is a proper class inner model of ZFC, delta>omega_1 is regular, (delta^+)^M = delta^+, and in V there is no proper class inner model with a Woodin cardinal, then K^M \| delta is universal for all mice in V of cardinality less than delta.Comment: Revised version, incorporating the referee's suggestion