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Brent Cody - One of the best experts on this subject based on the ideXlab platform.

  • forcing a κ like principle to hold at a weakly compact cardinal
    Annals of Pure and Applied Logic, 2021
    Co-Authors: Brent Cody, Victoria Gitman, Chris Lambiehanson
    Abstract:

    Abstract Hellsten [Hel03a] proved that when κ is Π n 1 -indescribable, the n-club subsets of κ provide a Filter Base for the Π n 1 -indescribability ideal, and hence can also be used to give a characterization of Π n 1 -indescribable sets which resembles the definition of stationarity: a set S ⊆ κ is Π n 1 -indescribable if and only if S ∩ C ≠ ∅ for every n-club C ⊆ κ . By replacing clubs with n-clubs in the definition of □ ( κ ) , one obtains a □ ( κ ) -like principle □ n ( κ ) , a version of which was first considered by Brickhill and Welch [BW] . The principle □ n ( κ ) is consistent with the Π n 1 -indescribability of κ but inconsistent with the Π n + 1 1 -indescribability of κ. By generalizing the standard forcing to add a □ ( κ ) -sequence, we show that if κ is κ + -weakly compact and GCH holds then there is a cofinality-preserving forcing extension in which κ remains κ + -weakly compact and □ 1 ( κ ) holds. If κ is Π 2 1 -indescribable and GCH holds then there is a cofinality-preserving forcing extension in which κ is κ + -weakly compact, □ 1 ( κ ) holds and every weakly compact subset of κ has a weakly compact proper initial segment. As an application, we prove that, relative to a Π 2 1 -indescribable cardinal, it is consistent that κ is κ + -weakly compact, every weakly compact subset of κ has a weakly compact proper initial segment, and there exist two weakly compact subsets S 0 and S 1 of κ such that there is no β κ for which both S 0 ∩ β and S 1 ∩ β are weakly compact.

  • characterizations of the weakly compact ideal on p κ λ
    Annals of Pure and Applied Logic, 2020
    Co-Authors: Brent Cody
    Abstract:

    Abstract Hellsten [15] gave a characterization of Π n 1 -indescribable subsets of a Π n 1 -indescribable cardinal in terms of a natural Filter Base: when κ is a Π n 1 -indescribable cardinal, a set S ⊆ κ is Π n 1 -indescribable if and only if S ∩ C ≠ ∅ for every n-club C ⊆ κ . We generalize Hellsten's characterization to Π n 1 -indescribable subsets of P κ λ , which were first defined by Baumgartner. First we show that under reasonable assumptions the Π 0 1 -indescribability ideal on P κ λ equals the minimal strongly normal ideal NSS κ , λ on P κ λ , which is different from NS κ , λ . We then formulate a notion of n-club subset of P κ λ and prove that a set S ⊆ P κ λ is Π n 1 -indescribable if and only if S ∩ C ≠ ∅ for every n-club C ⊆ P κ λ . We also prove that elementary embeddings considered by Schanker [26] witnessing near supercompactness lead to the definition of a normal ideal on P κ λ , and indeed, this ideal is equal to Baumgartner's ideal of non– Π 1 1 -indescribable subsets of P κ λ . Additionally, as applications of these results we answer a question of Cox-Lucke [10] about F -layered posets, provide a characterization of Π n m -indescribable subsets of P κ λ in terms of generic elementary embeddings and prove several results involving a two-cardinal weakly compact diamond principle.

  • Forcing a $\square(\kappa)$-like principle to hold at a weakly compact cardinal
    arXiv: Logic, 2019
    Co-Authors: Brent Cody, Victoria Gitman, Chris Lambie-hanson
    Abstract:

    Hellsten \cite{MR2026390} proved that when $\kappa$ is $\Pi^1_n$-indescribable, the \emph{$n$-club} subsets of $\kappa$ provide a Filter Base for the $\Pi^1_n$-indescribability ideal, and hence can also be used to give a characterization of $\Pi^1_n$-indescribable sets which resembles the definition of stationarity: a set $S\subseteq\kappa$ is $\Pi^1_n$-indescribable if and only if $S\cap C\neq\emptyset$ for every $n$-club $C\subseteq\kappa$. By replacing clubs with $n$-clubs in the definition of $\Box(\kappa)$, one obtains a $\Box(\kappa)$-like principle $\Box_n(\kappa)$, a version of which was first considered by Brickhill and Welch \cite{BrickhillWelch}. The principle $\Box_n(\kappa)$ is consistent with the $\Pi^1_n$-indescribability of $\kappa$ but inconsistent with the $\Pi^1_{n+1}$-indescribability of $\kappa$. By generalizing the standard forcing to add a $\Box(\kappa)$-sequence, we show that if $\kappa$ is $\kappa^+$-weakly compact and $\mathrm{GCH}$ holds then there is a cofinality-preserving forcing extension in which $\kappa$ remains $\kappa^+$-weakly compact and $\Box_1(\kappa)$ holds. If $\kappa$ is $\Pi^1_2$-indescribable and $\mathrm{GCH}$ holds then there is a cofinality-preserving forcing extension in which $\kappa$ is $\kappa^+$-weakly compact, $\Box_1(\kappa)$ holds and every weakly compact subset of $\kappa$ has a weakly compact proper initial segment. As an application, we prove that, relative to a $\Pi^1_2$-indescribable cardinal, it is consistent that $\kappa$ is $\kappa^+$-weakly compact, every weakly compact subset of $\kappa$ has a weakly compact proper initial segment, and there exist two weakly compact subsets $S^0$ and $S^1$ of $\kappa$ such that there is no $\beta

Chris Lambiehanson - One of the best experts on this subject based on the ideXlab platform.

  • forcing a κ like principle to hold at a weakly compact cardinal
    Annals of Pure and Applied Logic, 2021
    Co-Authors: Brent Cody, Victoria Gitman, Chris Lambiehanson
    Abstract:

    Abstract Hellsten [Hel03a] proved that when κ is Π n 1 -indescribable, the n-club subsets of κ provide a Filter Base for the Π n 1 -indescribability ideal, and hence can also be used to give a characterization of Π n 1 -indescribable sets which resembles the definition of stationarity: a set S ⊆ κ is Π n 1 -indescribable if and only if S ∩ C ≠ ∅ for every n-club C ⊆ κ . By replacing clubs with n-clubs in the definition of □ ( κ ) , one obtains a □ ( κ ) -like principle □ n ( κ ) , a version of which was first considered by Brickhill and Welch [BW] . The principle □ n ( κ ) is consistent with the Π n 1 -indescribability of κ but inconsistent with the Π n + 1 1 -indescribability of κ. By generalizing the standard forcing to add a □ ( κ ) -sequence, we show that if κ is κ + -weakly compact and GCH holds then there is a cofinality-preserving forcing extension in which κ remains κ + -weakly compact and □ 1 ( κ ) holds. If κ is Π 2 1 -indescribable and GCH holds then there is a cofinality-preserving forcing extension in which κ is κ + -weakly compact, □ 1 ( κ ) holds and every weakly compact subset of κ has a weakly compact proper initial segment. As an application, we prove that, relative to a Π 2 1 -indescribable cardinal, it is consistent that κ is κ + -weakly compact, every weakly compact subset of κ has a weakly compact proper initial segment, and there exist two weakly compact subsets S 0 and S 1 of κ such that there is no β κ for which both S 0 ∩ β and S 1 ∩ β are weakly compact.

Roy R. - One of the best experts on this subject based on the ideXlab platform.

  • Ultrasound Image Filtering and Reconstruction Using DCT/IDCT Filter Structure
    'Institute of Electrical and Electronics Engineers (IEEE)', 2020
    Co-Authors: Honarvar Shakibaei Asli B., Flusser J., Zhao Y., Erkoyuncu J. A., Banerjee Krishnan K., Farrokhi Y., Roy R.
    Abstract:

    In this paper, a new recursive structure Based on the convolution model of discrete cosine transform (DCT) for designing of a finite impulse response (FIR) digital Filter is proposed. In our derivation, we start with the convolution model of DCT-II to use its Z-transform for the proposed Filter structure perspective. Moreover, using the same algorithm, a Filter Base implementation of the inverse DCT (IDCT) for image reconstruction is developed. The computational time experiments of the proposed DCT/IDCT Filter(s) demonstrate that the proposed Filters achieve faster elapsed CPU time compared to the direct recursive structures and recursive algorithms for the DCT/IDCT with Arbitrary Length. Experimental results on clinical ultrasound images and comparisons with classical Wiener Filter, non-local mean (NLM) Filter and total variation (TV) algorithms are used to validate the improvements of the proposed approaches in both noise reduction and reconstruction performance for ultrasound images

Roy Rajkumar - One of the best experts on this subject based on the ideXlab platform.

  • DCT/IDCT Filter design for ultrasound image Filtering
    'Institute of Electrical and Electronics Engineers (IEEE)', 2021
    Co-Authors: Shakibaei, Barmak Honarvar, Flusser Jan, Zhao Yifan, Erkoyuncu, John Ahmet, Roy Rajkumar
    Abstract:

    In this paper, a new recursive structure Based on the convolution model of discrete cosine transform (DCT) for designing of a finite impulse response (FIR) digital Filter is proposed. In our derivation, we start with the convolution model of DCT-II to use its Z-transform for the proposed Filter structure perspective. Moreover, using the same algorithm, a Filter Base implementation of the inverse DCT (IDCT) for image reconstruction is developed. The computational time experiments of the proposed DCT/IDCT Filter(s) demonstrate that the proposed Filters achieve faster elapsed CPU time compared to the others. The image Filtering and reconstruction performance of the proposed approach on ultrasound images are presented to validate the theoretical framework

Victoria Gitman - One of the best experts on this subject based on the ideXlab platform.

  • forcing a κ like principle to hold at a weakly compact cardinal
    Annals of Pure and Applied Logic, 2021
    Co-Authors: Brent Cody, Victoria Gitman, Chris Lambiehanson
    Abstract:

    Abstract Hellsten [Hel03a] proved that when κ is Π n 1 -indescribable, the n-club subsets of κ provide a Filter Base for the Π n 1 -indescribability ideal, and hence can also be used to give a characterization of Π n 1 -indescribable sets which resembles the definition of stationarity: a set S ⊆ κ is Π n 1 -indescribable if and only if S ∩ C ≠ ∅ for every n-club C ⊆ κ . By replacing clubs with n-clubs in the definition of □ ( κ ) , one obtains a □ ( κ ) -like principle □ n ( κ ) , a version of which was first considered by Brickhill and Welch [BW] . The principle □ n ( κ ) is consistent with the Π n 1 -indescribability of κ but inconsistent with the Π n + 1 1 -indescribability of κ. By generalizing the standard forcing to add a □ ( κ ) -sequence, we show that if κ is κ + -weakly compact and GCH holds then there is a cofinality-preserving forcing extension in which κ remains κ + -weakly compact and □ 1 ( κ ) holds. If κ is Π 2 1 -indescribable and GCH holds then there is a cofinality-preserving forcing extension in which κ is κ + -weakly compact, □ 1 ( κ ) holds and every weakly compact subset of κ has a weakly compact proper initial segment. As an application, we prove that, relative to a Π 2 1 -indescribable cardinal, it is consistent that κ is κ + -weakly compact, every weakly compact subset of κ has a weakly compact proper initial segment, and there exist two weakly compact subsets S 0 and S 1 of κ such that there is no β κ for which both S 0 ∩ β and S 1 ∩ β are weakly compact.

  • Forcing a $\square(\kappa)$-like principle to hold at a weakly compact cardinal
    arXiv: Logic, 2019
    Co-Authors: Brent Cody, Victoria Gitman, Chris Lambie-hanson
    Abstract:

    Hellsten \cite{MR2026390} proved that when $\kappa$ is $\Pi^1_n$-indescribable, the \emph{$n$-club} subsets of $\kappa$ provide a Filter Base for the $\Pi^1_n$-indescribability ideal, and hence can also be used to give a characterization of $\Pi^1_n$-indescribable sets which resembles the definition of stationarity: a set $S\subseteq\kappa$ is $\Pi^1_n$-indescribable if and only if $S\cap C\neq\emptyset$ for every $n$-club $C\subseteq\kappa$. By replacing clubs with $n$-clubs in the definition of $\Box(\kappa)$, one obtains a $\Box(\kappa)$-like principle $\Box_n(\kappa)$, a version of which was first considered by Brickhill and Welch \cite{BrickhillWelch}. The principle $\Box_n(\kappa)$ is consistent with the $\Pi^1_n$-indescribability of $\kappa$ but inconsistent with the $\Pi^1_{n+1}$-indescribability of $\kappa$. By generalizing the standard forcing to add a $\Box(\kappa)$-sequence, we show that if $\kappa$ is $\kappa^+$-weakly compact and $\mathrm{GCH}$ holds then there is a cofinality-preserving forcing extension in which $\kappa$ remains $\kappa^+$-weakly compact and $\Box_1(\kappa)$ holds. If $\kappa$ is $\Pi^1_2$-indescribable and $\mathrm{GCH}$ holds then there is a cofinality-preserving forcing extension in which $\kappa$ is $\kappa^+$-weakly compact, $\Box_1(\kappa)$ holds and every weakly compact subset of $\kappa$ has a weakly compact proper initial segment. As an application, we prove that, relative to a $\Pi^1_2$-indescribable cardinal, it is consistent that $\kappa$ is $\kappa^+$-weakly compact, every weakly compact subset of $\kappa$ has a weakly compact proper initial segment, and there exist two weakly compact subsets $S^0$ and $S^1$ of $\kappa$ such that there is no $\beta