Many-times huge and superhuge cardinals

1984 ◽  
Vol 49 (1) ◽  
pp. 112-122 ◽  
Author(s):  
Julius B. Barbanel ◽  
Carlos A. Diprisco ◽  
It Beng Tan
Keyword(s):  
New York ◽  
Positive Integer ◽  
Critical Point ◽  
Order Type ◽  
Elementary Embedding ◽  
Rochester Institute Of Technology ◽  
Inner Model ◽  
Institute Of Technology ◽  
The Universe ◽  
Normal Ultrafilter

In this paper we consider various generalizations of the notion of hugeness. We remind the reader that a cardinal κ is huge if there exist a cardinal λ > κ, an inner model M which is closed under λ-sequences, and an elementary embedding i: V → M with critical point κ such that i(κ) = λ. We shall call λ a target for κ and shall write κ → (λ) to express this fact. Equivalently, κ is huge with target λ if and only if there exists a normal ultrafilter on P=κ(λ) = {X ⊆ λ:X has order type κ}. For the proof and additional facts on hugeness, see [3].We assume that the reader is familiar with the notions of measurability and supercompactness. If κ is γ-supercompact for each γ < λ, we shall say that κ is < λ-supercompact. We note that if κ → (λ), it follows immediately that κ is < λ-supercompact.Throughout the paper, n shall be used to denote a positive integer, the letters α, β, and δ shall denote ordinals, while κ, λ, γ, and η shall be reserved for cardinals. All addition is ordinal addition. V denotes the universe of all sets.All results except for Theorems 6b and 6c and Lemma 6d can be formalized in ZFC.This paper was written while the first named author was at Rochester Institute of Technology, Rochester, New York. We wish to thank the department of mathematics at R.I.T. for secretarial time and facilities.

Strongly compact cardinals, elementary embeddings and fixed points

1984 ◽  
Vol 49 (3) ◽  
pp. 808-812
Author(s):  
Yoshihiro Abe
Keyword(s):  
Fixed Points ◽  
Elementary Embedding ◽  
Inner Model ◽  
Power Set ◽  
Elementary Embeddings ◽  
Basic Facts ◽  
The Universe ◽  
Normal Ultrafilter ◽  
Strongly Compact Cardinals

J. Barbanel [1] characterized the class of cardinals fixed by an elementary embedding induced by a normal ultrafilter on Pκλ assuming that κ is supercompact. In this paper we shall prove the same results from the weaker hypothesis that κ is strongly compact and the ultrafilter is fine.We work in ZFC throughout. Our set-theoretic notation is quite standard. In particular, if X is a set, ∣X∣ denotes the cardinality of X and P(X) denotes the power set of X. Greek letters will denote ordinals. In particular γ, κ, η and γ will denote cardinals. If κ and λ are cardinals, then λ<κ is defined to be supγ<κγγ. Cardinal exponentiation is always associated from the top. Thus, for example, 2λ<κ means 2(λ<κ). V denotes the universe of all sets. If M is an inner model of ZFC, ∣X∣M and P(X)M denote the cardinality of X in M and the power set of X in M respectively.We review the basic facts on fine ultrafilters and the corresponding elementary embeddings. (For detail, see [2].)Definition. Assume κ and λ are cardinals with κ ≤ λ. Then, Pκλ = {X ⊂ λ∣∣X∣ < κ}.It is important to note that ∣Pκλ∣ = λ< κ.

On P-points over a measurable cardinal

1981 ◽  
Vol 46 (1) ◽  
pp. 59-66
Author(s):  
A. Kanamori
Keyword(s):  
Measurable Cardinal ◽  
Focus Of Attention ◽  
Large Cardinal ◽  
Elementary Embedding ◽  
Inner Model ◽  
Measurable Cardinals ◽  
The Universe

This paper continues the study of κ-ultrafilters over a measurable cardinal κ, following the sequence of papers Ketonen [2], Kanamori [1] and Menas [4]. Much of the concern will be with p-point κ-ultrafilters, which have become a focus of attention because they epitomize situations of further complexity beyond the better understood cases, normal and product κ-ultrafilters.For any κ-ultrafilter D, let iD: V → MD ≃ Vκ/D be the elementary embedding of the universe into the transitization of the ultrapower by D. Situations of U < RKD will be exhibited when iU(κ) < iD(κ), and when iU(κ) = iD(κ). The main result will then be that if the latter case obtains, then there is an inner model with two measurable cardinals. (As will be pointed out, this formulation is due to Kunen, and improves on an earlier version of the author.) Incidentally, a similar conclusion will also follow from the assertion that there is an ascending Rudin-Keisler chain of κ-ultrafilters of length ω + 1. The interest in these results lies in the derivability of a substantial large cardinal assertion from plausible hypotheses on κ-ultrafilters.

Lahham, Rafiq (1932--)

2018 ◽  
Author(s):  
Sarah Ann Rogers
Keyword(s):  
New York ◽  
Subject Matter ◽  
First Generation ◽  
Modern Art ◽  
Founding Member ◽  
Rochester Institute Of Technology ◽  
Institute Of Technology ◽  
The 1960S ◽  
To Receive ◽  
Arabic Calligraphy

Born in Damascus in 1932, Rafiq Lahham went on to become a pioneer in Jordan’s modern art movement. His body of work is characterized by a diverse approach to choice of style, media, and subject matter. Working in oil, gouache, watercolor, collage, printmaking, and silk screens, Lahham depicts portraits, landscapes and cityscapes, Arabic calligraphy, and semi-abstract compositions. During the 1960s, Lahham was among the first artists in Jordan to incorporate calligraphy into his compositions and also one of the first painters to experiment with complete abstraction. Lahham is considered, along with Muhanna Durra, to be a member of the first generation of Jordanian artists to receive government scholarships to train abroad. He studied at Ente Nationale Addestramento Lavoratori Commercio and St Giacomo Instituto in Rome, graduating in 1962. He continued his studies in painting and etching at the Rochester Institute of Technology in New York. Upon his return to Amman, Lahham worked as a cultural advisor for the Ministry of Tourism until his retirement in 1995. He is a founding member of the Artists Association. He lives and works in Amman.

Estimating the Biogas Potential From Colleges and Universities

2014 ◽  
Author(s):  
Jacqueline Ebner ◽  
Shwe Sin Win ◽  
Swati Hegde ◽  
Scott Vadney ◽  
Anahita Williamson ◽  
...  
Keyword(s):  
New York ◽  
Food Waste ◽  
Electrical Power ◽  
New York State ◽  
Colleges And Universities ◽  
Dairy Manure ◽  
Top Down ◽  
Bottom Up ◽  
Rochester Institute Of Technology ◽  
Institute Of Technology

Academic institutions present a unique opportunity for anaerobic digestion (AD) projects in that they have a concentrated population that generates waste, utilizes heat and electrical power, and often are motivated to implement sustainability initiatives. However, implementation of AD on college campuses in the U.S. is only beginning to emerge and data required to size and operate digesters are limited. This paper provides formulae to estimate food waste generated at college and university campuses base upon data collected at the Rochester Institute of Technology. Bottom-up and top-down estimates are presented and results are compared to an extensive review of publicly available data from other colleges and universities. The bottom-up methodology resulted in a lower estimate (18 kg food waste/enrolled student) than the top-down estimate (29 kg/enrolled student). Both were significantly lower than the estimate previously reported in the literature (64 kg/enrolled student). Bench-scale co-digestion experiments of the food waste with dairy manure resulted in a methane yield of 437 ml CH4/g VS. Applying this methodology to only 4-year colleges in New York State has the potential to generate 27 million GJ of energy from food waste.

Herstory of Graphic Design: Elaine Lustig Cohen

Collections ◽  
2019 ◽  
Vol 15 (2-3) ◽  
pp. 167-177
Author(s):  
Megan Moltrup
Keyword(s):  
New York ◽  
Graphic Design ◽  
Design History ◽  
Graphic Arts ◽  
Representation Of Women ◽  
Rochester Institute Of Technology ◽  
History Of ◽  
Institute Of Technology ◽  
To Receive ◽  
The Relationship

The Cary Graphic Arts Collection in Rochester, New York, manages the Graphic Design Archive of the Rochester Institute of Technology which features more than 35 collections documenting the work of many 20th-century Modernist graphic designers. Among these is the work of Elaine Lustig Cohen (1927–2016), a relatively unknown designer from New York City. Upon her marriage to the well-known designer Alvin Lustig, Elaine unknowingly started out on her path as a designer. She seamlessly transitioned from office manager to artist, but it took decades for her to receive recognition for her work. In an attempt to situate Elaine Lustig Cohen and her body of work within graphic design history and to give her body of work greater attention, I researched, handled, and disseminated knowledge of her work and her collection. Specifically, I examined and organized her collection at the Cary Graphic Arts Collection and went on to co-curate an exhibition chronicling her career as part of my capstone of my undergraduate degree in museum studies. I wanted to look at this collection in relation to the bigger picture of women in design and to the relationship between the representation of women in the history of graphic design textbooks and the availability of their work in archives.

scholarly journals Whole-Genome Sequencing and Annotation of 10 Endophytic and Epiphytic Bacteria Isolated from Lolium arundinaceum

2021 ◽  
Vol 10 (19) ◽  
Author(s):  
Victoria A. Gaeth ◽  
Christina J. Domondon ◽  
Paul A. Podbielski ◽  
Virginia X. Aswad ◽  
Emalee A. Wrightstone ◽  
...  
Keyword(s):  
New York ◽  
Whole Genome Sequence ◽  
Whole Genome ◽  
Epiphytic Bacteria ◽  
Lolium Arundinaceum ◽  
Content Type ◽  
Laboratory Exercise ◽  
Rochester Institute Of Technology ◽  
Plant Biochemistry ◽  
Institute Of Technology

We report the whole-genome sequence and annotation of 10 endophytic and epiphytic bacteria isolated from the grass Lolium arundinaceum as part of a laboratory exercise in a Fundamentals of Plant Biochemistry and Pathology undergraduate course (BIOL403) at the Rochester Institute of Technology in Rochester, New York.

Co-critical points of elementary embeddings

1985 ◽  
Vol 50 (1) ◽  
pp. 220-226
Author(s):  
Michael Sheard
Keyword(s):  
Critical Point ◽  
Measurable Cardinal ◽  
Elementary Embedding ◽  
Proper Class ◽  
Ground Model ◽  
Target Model ◽  
Inner Models ◽  
Inner Model ◽  
Elementary Embeddings

Probably the two most famous examples of elementary embeddings between inner models of set theory are the embeddings of the universe into an inner model given by a measurable cardinal and the embeddings of the constructible universeLinto itself given by 0#. In both of these examples, the “target model” is a subclass of the “ground model” (and in the latter case they are equal). It is not hard to find examples of embeddings in which the target model is not a subclass of the ground model: ifis a generic ultrafilter arising from forcing with a precipitous ideal on a successor cardinalκ, then the ultraproduct of the ground model viacollapsesκ. Such considerations suggest a classification of how close the target model comes to “fitting inside” the ground model.Definition 1.1. LetMandNbe inner models (transitive, proper class models) of ZFC, and letj:M→Nbe an elementary embedding. Theco-critical pointofjis the least ordinalλ, if any exist, such that there isX⊆λ, X∈NbutX∉M. Such anXis called anew subsetofλ.It is easy to see that the co-critical point ofj:M→Nis a cardinal inN.

On elementary embeddings from an inner model to the universe

2001 ◽  
Vol 66 (3) ◽  
pp. 1090-1116 ◽  
Author(s):  
J. Vickers ◽  
P. D. Welch
Keyword(s):  
Measurable Cardinal ◽  
Negative Answer ◽  
Core Model ◽  
Elementary Embedding ◽  
Proper Class ◽  
The Core ◽  
Inner Model ◽  
Elementary Embeddings ◽  
The Universe ◽  
Closure Assumption

AbstractWe consider the following question of Kunen:Does Con(ZFC + ∃M a transitive inner model and a non-trivial elementary embedding j: M → V)imply Con(ZFC + ∃ a measurable cardinal)?We use core model theory to investigate consequences of the existence of such a j: M → V. We prove, amongst other things, the existence of such an embedding implies that the core model K is a model of “there exists a proper class of almost Ramsey cardinals”. Conversely, if On is Ramsey, then such a j. M are definable.We construe this as a negative answer to the question above. We consider further the consequences of strengthening the closure assumption on j to having various classes of fixed points.

scholarly journals The Kelmscott/Goudy Printing Press: Printing History as a Living History

2016 ◽  
Vol 17 (1) ◽  
pp. 15-16
Author(s):  
Steven K. Galbraith
Keyword(s):  
New York ◽  
New York City ◽  
York City ◽  
Printing Press ◽  
Book Arts ◽  
Living History ◽  
Graphic Arts ◽  
Rochester Institute Of Technology ◽  
The Press ◽  
Institute Of Technology

The arrival of the Kelmscott/Goudy press to the Cary Graphic Arts Collection at the Rochester Institute of Technology in January 2014 was a homecoming of sorts. From 1932 to 1941, the press belonged to our library’s namesake, Melbert B. Cary, Jr., director of the Continental Type Founders Association in New York City. Cary used the press to produce the whimsical publications of his Press of the Woolly Whale. In addition to its connection with the press’s past, the Cary Collection offers a home where the press can be maintained and used in support of teaching and the book arts. To . . .

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