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1. Quadratic rotation symmetric Boolean functions

Author

Chirvasitu, Alexandru and Cusick, Thomas W.

Subjects
Mathematics - Combinatorics, Computer Science - Information Theory, Mathematics - Number Theory, 94A60, 06E30, 94C99, 11R18, 11T06, 11R29, 15B34, 15A18
Abstract

Let $(0, a_1, \ldots, a_{d-1})_n$ denote the function $f_n(x_0, x_1, \ldots, x_{n-1})$ of degree $d$ in $n$ variables generated by the monomial $x_0x_{a_1} \cdots x_{a_{d-1}}$ and having the property that $f_n$ is invariant under cyclic permutations of the variables. Such a function $f_n$ is called monomial rotation symmetric (MRS). Much of this paper extends the work on quadratic MRS functions in a $2020$ paper of the authors to the case of binomial RS functions, that is sums of two quadratic MRS functions. There are also some results for the sum of any number of quadratic MRS functions., Comment: 20 pages + references

Published
2023

3. Comparison of outcomes of different biopsy schedules among men on active surveillance for prostate cancer: An analysis of the G.A.P.3 global consortium database

Author

Beckmann, Kerri R, Bangma, Chris H, Helleman, Jozien, Bjartell, Anders, Carroll, Peter R, Morgan, Todd, Nieboer, Daan, Santaolalla, Aida, Trock, Bruce J, Valdagni, Riccardo, Roobol, Monique J, Trock, Bruce, Ehdaie, Behfar, Carroll, Peter, Filson, Christopher, Logothetis, Christopher, Klotz, Laurence, Pickles, Tom, Hyndman, Eric, Moore, Caroline, Gnanapragasam, Vincent, Van Hemelrijck, Mieke, Dasgupta, Prokar, Bangma, Chris, Roobol, Monique, Villers, Arnauld, Robert, Grégoire, Semjonow, Axel, Rannikko, Antti, Perry, Antoinette, Hugosson, Jonas, Rubio‐Briones, Jose, Hefermehl, Lukas, Shiong, Lee Lui, Frydenberg, Mark, Stricker, Phillip, Sugimoto, Mikio, Chung, Byung Ha, van der Kwast, Theo, van der Linden, Wim, Hulsen, Tim, Ruwe, Boris, van Hooft, Peter, Steyerberg, Ewout, Beckmann, Kerri, Denton, Brian, Hayen, Andrew, Boutros, Paul, Guo, Wei, Benfante, Nicole, Cowan, Janet, Patil, Dattatraya, Park, Lauren, Ferrante, Stephanie, Mamedov, Alexandre, LaPointe, Vincent, Crump, Trafford, Stavrinides, Vasilis, Kimberly‐Duffell, Jenna, Olivier, Jonathan, Rancati, Tiziana, Ahlgren, Helén, Mascarós, Juanma, Löfgren, Annica, Lehmann, Kurt, Lin, Catherine Han, Cusick, Thomas, Hirama, Hiromi, Lee, Kwang Suk, Jenster, Guido, Auvinen, Anssi, Haider, Masoom, van Bochove, Kees, Kouspou, Michelle, and Paich, Kellie

Subjects
Biomedical and Clinical Sciences, Clinical Sciences, Oncology and Carcinogenesis, Urologic Diseases, Aging, Prevention, Cancer, Prostate Cancer, Clinical Research, Biopsy, Disease Progression, Humans, Male, Neoplasm Grading, Prostate, Prostate-Specific Antigen, Prostatic Neoplasms, Watchful Waiting, active surveillance, biopsy schedule, prostate cancer, treatment, upgrading, Global Action Plan Active Surveillance Prostate Cancer [G.A.P.3] Consortium, Paediatrics and Reproductive Medicine, Oncology & Carcinogenesis, Clinical sciences, Oncology and carcinogenesis
Abstract

BackgroundThe optimal interval for repeat biopsy during active surveillance (AS) for prostate cancer is yet to be defined. This study examined whether risk of upgrading (to grade group ≥ 2) or risk of converting to treatment varied according to intensity of repeat biopsy using data from the GAP3 consortium's global AS database.Materials and methodsIntensity of surveillance biopsy schedules was categorized according to centers' protocols: (a) Prostate Cancer Research International Active Surveillance project (PRIAS) protocols with biopsies at years 1, 4, and 7 (10 centers; 7532 men); (b) biennial biopsies, that is, every other year (8 centers; 4365 men); and (c) annual biopsy schedules (4 centers; 1602 men). Multivariable Cox regression was used to compare outcomes according to biopsy intensity.ResultsOut of the 13,508 eligible participants, 56% were managed according to PRIAS protocols (biopsies at years 1, 4, and 7), 32% via biennial biopsy, and 12% via annual biopsy. After adjusting for baseline characteristics, risk of converting to treatment was greater for those on annual compared with PRIAS biopsy schedules (hazard ratio [HR] = 1.66; 95% confidence interval [CI] = 1.51-1.83; p

Published
2022

4. Prostate Cancer Patients Under Active Surveillance with a Suspicious Magnetic Resonance Imaging Finding Are at Increased Risk of Needing Treatment: Results of the Movember Foundation’s Global Action Plan Prostate Cancer Active Surveillance (GAP3) Consortium

Author

Olivier, Jonathan, Li, Weiyu, Nieboer, Daan, Helleman, Jozien, Roobol, Monique, Gnanapragasam, Vincent, Frydenberg, Mark, Sugimoto, Mikio, Carroll, Peter, Morgan, Todd M, Valdagni, Riccardo, Rubio-Briones, Jose, Robert, Grégoire, Stricker, Phillip, Hayen, Andrew, Schoots, Ivo, Haider, Masoom, Moore, Caroline M, Denton, Brian, Villers, Arnauld, Trock, Bruce, Ehdaie, Behfar, Filson, Christopher, Logothetis, Christopher, Morgan, Todd, Klotz, Laurence, Pickles, Tom, Hyndman, Eric, Van Hemelrijck, Mieke, Dasgupta, Prokar, Bangma, Chris, Semjonow, Axel, Rannikko, Antti, Perry, Antoinette, Hugosson, Jonas, Bjartell, Anders, Hefermehl, Lukas, Shiong, Lee Lui, Chung, Byung Ha, van der Kwast, Theo, Hulsen, Tim, van der Linden, Wim, Ruwe, Boris, van Hooft, Peter, Steyerberg, Ewout, Boutros, Paul, Guo, Wei, Benfante, Nicole, Cowan, Janet, Patil, Dattatraya, Park, Lauren, Ferrante, Stephanie, Mamedov, Alexandre, LaPointe, Vincent, Crump, Trafford, Stavrinides, Vasilis, Kimberly-Duffell, Jenna, Santaolalla, Aida, Rancati, Tiziana, Ahlgren, Helén, Mascarós, Juanma, Löfgren, Annica, Lehmann, Kurt, Lin, Catherine Han, Cusick, Thomas, Hirama, Hiromi, Lee, Kwang Suk, Jenster, Guido, Auvinen, Anssi, van Bochove, Kees, Kouspou, Michelle, and Paich, Kellie

Subjects
Biomedical and Clinical Sciences, Clinical Sciences, Oncology and Carcinogenesis, Prevention, Clinical Research, Biomedical Imaging, Clinical Trials and Supportive Activities, Aging, Cancer, Urologic Diseases, Prostate Cancer, Detection, screening and diagnosis, 4.2 Evaluation of markers and technologies, Magnetic resonance imaging, Prostate cancer, Active surveillance, Discontinuation, Worldwide, Nomogram, Movember Foundation's Global Action Plan Prostate Cancer Active Surveillance GAP3 Consortium, Urology & Nephrology, Clinical sciences
Abstract

BackgroundThe inclusion criterion for active surveillance (AS) is low- or intermediate-risk prostate cancer. The predictive value of the presence of a suspicious lesion at magnetic resonance imaging (MRI) at the time of inclusion is insufficiently known.ObjectiveTo evaluate the percentage of patients needing active treatment stratified by the presence or absence of a suspicious lesion at baseline MRI.Design setting and participantsA retrospective analysis of the data from the multicentric AS GAP3 Consortium database was conducted. The inclusion criteria were men with grade group (GG) 1 or GG 2 prostate cancer combined with prostate-specific antigen

Published
2022

7. Symbolic dynamics and rotation symmetric Boolean functions

Author

Chirvasitu, Alexandru and Cusick, Thomas

Subjects
Computer Science - Information Theory, Mathematics - Dynamical Systems, Mathematics - Number Theory, 06E30, 37B50, 11G20
Abstract

We identify the weights $wt(f_n)$ of a family $\{f_n\}$ of rotation symmetric Boolean functions with the cardinalities of the sets of $n$-periodic points of a finite-type shift, recovering the second author's result that said weights satisfy a linear recurrence. Similarly, the weights of idempotent functions $f_n$ defined on finite fields can be recovered as the cardinalities of curves over those fields and hence satisfy a linear recurrence as a consequence of the rationality of curves' zeta functions. Weil's Riemann hypothesis for curves then provides additional information about $wt(f_n)$. We apply our results to the case of quadratic functions and considerably extend the results in an earlier paper of ours., Comment: 23 pages + references

Published
2019

8. Affine equivalence for quadratic rotation symmetric Boolean functions

Author

Chirvasitu, Alexandru and Cusick, Thomas W.

Subjects
Computer Science - Information Theory, Mathematics - Rings and Algebras, 06E30
Abstract

Let $f_n(x_0, x_1, \ldots, x_{n-1})$ denote the algebraic normal form (polynomial form) of a rotation symmetric (RS) Boolean function of degree $d$ in $n \geq d$ variables and let $wt(f_n)$ denote the Hamming weight of this function. Let $(0, a_1, \ldots, a_{d-1})_n$ denote the function $f_n$ of degree $d$ in $n$ variables generated by the monomial $x_0x_{a_1} \cdots x_{a_{d-1}}.$ Such a function $f_n$ is called monomial rotation symmetric (MRS). It was proved in a $2012$ paper that for any MRS $f_n$ with $d=3,$ the sequence of weights $\{w_k = wt(f_k):~k = 3, 4, \ldots\}$ satisfies a homogeneous linear recursion with integer coefficients. This result was gradually generalized in the following years, culminating around $2016$ with the proof that such recursions exist for any rotation symmetric function $f_n.$ Recursions for quadratic RS functions were not explicitly considered, since a $2009$ paper had already shown that the quadratic weights themselves could be given by an explicit formula. However, this formula is not easy to compute for a typical quadratic function. This paper shows that the weight recursions for the quadratic RS functions have an interesting special form which can be exploited to solve various problems about these functions, for example, deciding exactly which quadratic RS functions are balanced., Comment: 27 pages + references; minor typo corrections

Published
2019

10. Simpler proof for nonlinearity of majority function

Author

Cusick, Thomas W.

Subjects
Computer Science - Information Theory, 94A60 06E30
Abstract

Given a Boolean function f, the (Hamming) weight wt(f) and the nonlinearity N(f) are well known to be important in designing functions that are useful in cryptography. The nonlinearity is expensive to compute, in general, so any shortcuts for doing that for particular functions f are significant. The well known majority function has been extensively studied in a cryptographic context for the last dozen years or so, and there is a formula for its nonlinearity. The known proofs for this formula rely on many detailed results for the Krawtchouk polynomials. This paper gives a much simpler proof., Comment: 10 pages, LaTeX; better title and abstract; reference added; revised proofs

Published
2017

11. Active Surveillance for Men Younger than 60 Years or with Intermediate-risk Localized Prostate Cancer. Descriptive Analyses of Clinical Practice in the Movember GAP3 Initiative

Author

Trock, Bruce, Ehdaie, Behfar, Carroll, Peter, Filson, Christopher, Logothetis, Christopher, Morgan, Todd, Klotz, Laurence, Pickles, Tom, Hyndman, Eric, Moore, Caroline, Gnanapragasam, Vincent, Van Hemelrijck, Mieke, Dasgupta, Prokar, Bangma, Chris, Roobol, Monique, Villers, Arnauld, Robert, Grégoire, Semjonow, Axel, Rannikko, Antti, Valdagni, Riccardo, Perry, Antoinette, Hugosson, Jonas, Rubio-Briones, Jose, Bjartell, Anders, Hefermehl, Lukas, Lui Shiong, Lee, Frydenberg, Mark, Stricker, Phillip, Sugimoto, Mikio, Ha Chung, Byung, van der Kwast, Theo, van der Linden, Wim, Hulsen, Tim, Ruwe, Boris, Peter van Hooft, Steyerberg, Ewout, Nieboer, Daan, Beckmann, Kerri, Denton, Brian, Hayen, Andrew, Boutros, Paul, Guo, Wei, Benfante, Nicole, Cowan, Janet, Patil, Dattatraya, Park, Lauren, Ferrante, Stephanie, Mamedov, Alexandre, LaPointe, Vincent, Crump, Trafford, Stavrinides, Vasilis, Kimberly-Duffell, Jenna, Santaolalla, Aida, Olivier, Jonathan, Rancati, Tiziana, Ahlgren, Helén, Mascarós, Juanma, Löfgren, Annica, Lehmann, Kurt, Han Lin, Catherine, Cusick, Thomas, Hirama, Hiromi, Suk Lee, Kwang, Jenster, Guido, Auvinen, Anssi, Haider, Masoom, van Bochove, Kees, Kouspou, Michelle, Paich, Kellie, Helleman, Jozien, Remmers, Sebastiaan, Hyndman, Matthew E., Moore, Caroline M., Shiong Lee, Lui, Elhage, Oussama, Morgan, Todd M., Bangma, Chris H., and Roobol, Monique J.

Published
2022
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12. Prostate Cancer Patients Under Active Surveillance with a Suspicious Magnetic Resonance Imaging Finding Are at Increased Risk of Needing Treatment: Results of the Movember Foundation’s Global Action Plan Prostate Cancer Active Surveillance (GAP3) Consortium

Author

Trock, Bruce, Ehdaie, Behfar, Carroll, Peter, Filson, Christopher, Logothetis, Christopher, Morgan, Todd, Klotz, Laurence, Pickles, Tom, Hyndman, Eric, Moore, Caroline M, Gnanapragasam, Vincent, Van Hemelrijck, Mieke, Dasgupta, Prokar, Bangma, Chris, Roobol, Monique, Villers, Arnauld, Robert, Grégoire, Semjonow, Axel, Rannikko, Antti, Valdagni, Riccardo, Perry, Antoinette, Hugosson, Jonas, Rubio-Briones, Jose, Bjartell, Anders, Hefermehl, Lukas, Lui Shiong, Lee, Frydenberg, Mark, Stricker, Phillip, Sugimoto, Mikio, Ha Chung, Byung, van der Kwast, Theo, Hulsen, Tim, van der Linden, Wim, Ruwe, Boris, van Hooft, Peter, Steyerberg, Ewout, Nieboer, Daan, Denton, Brian, Hayen, Andrew, Boutros, Paul, Guo, Wei, Benfante, Nicole, Cowan, Janet, Patil, Dattatraya, Park, Lauren, Ferrante, Stephanie, Mamedov, Alexandre, LaPointe, Vincent, Crump, Trafford, Stavrinides, Vasilis, Kimberly-Duffell, Jenna, Santaolalla, Aida, Olivier, Jonathan, Rancati, Tiziana, Ahlgren, Helén, Mascarós, Juanma, Löfgren, Annica, Lehmann, Kurt, Han Lin, Catherine, Cusick, Thomas, Hirama, Hiromi, Suk Lee, Kwang, Jenster, Guido, Auvinen, Anssi, Haider, Masoom, van Bochove, Kees, Kouspou, Michelle, Paich, Kellie, Helleman, Jozien, Li, Weiyu, Morgan, Todd M., Schoots, Ivo, and Moore, Caroline M.

Published
2022
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13. Weight recursions for any rotation symmetric Boolean functions

Author

Cusick, Thomas W.

Subjects
Mathematics - Combinatorics, Computer Science - Information Theory, 94C10, 06E30, 94A60
Abstract

Let $f_n(x_1, x_2, \ldots, x_n)$ denote the algebraic normal form (polynomial form) of a rotation symmetric Boolean function of degree $d$ in $n \geq d$ variables and let $wt(f_n)$ denote the Hamming weight of this function. Let $(1, a_2, \ldots, a_d)_n$ denote the function $f_n$ of degree $d$ in $n$ variables generated by the monomial $x_1x_{a_2} \cdots x_{a_d}.$ Such a function $f_n$ is called {\em monomial rotation symmetric} (MRS). It was proved in a $2012$ paper that for any MRS $f_n$ with $d=3,$ the sequence of weights $\{w_k = wt(f_k):~k = 3, 4, \ldots\}$ satisfies a homogeneous linear recursion with integer coefficients. In this paper it is proved that such recursions exist for any rotation symmetric function $f_n;$ such a function is generated by some sum of $t$ monomials of various degrees. The last section of the paper gives a Mathematica program which explicitly computes the homogeneous linear recursion for the weights, given any rotation symmetric $f_n.$ The reader who is only interested in finding some recursions can use the program and not be concerned with the details of the rather complicated proofs in this paper., Comment: 18 pages

Published
2017

14. Comparison of Characteristics, Follow-up and Outcomes of Active Surveillance for Prostate Cancer According to Ethnicity in the GAP3 Global Consortium Database

Author

Trock, Bruce, Ehdaie, Behfar, Carroll, Peter, Filson, Christopher, Logothetis, Christopher, Morgan, Todd, Klotz, Laurence, Pickles, Tom, Hyndman, Eric, Moore, Caroline, Gnanapragasam, Vincent, Van Hemelrijck, Mieke, Dasgupta, Prokar, Bangma, Chris, Roobol, Monique, Villers, Arnauld, Robert, Grégoire, Semjonow, Axel, Rannikko, Antti, Valdagni, Riccardo, Perry, Antoinette, Hugosson, Jonas, Rubio-Briones, Jose, Bjartell, Anders, Hefermehl, Lukas, Lui Shiong, Lee, Frydenberg, Mark, Stricker, Phillip, Sugimoto, Mikio, Ha Chung, Byung, van der Kwast, Theo, van der Linden, Wim, Hulsen, Tim, Ruwe, Boris, van Hooft, Peter, Steyerberg, Ewout, Nieboer, Daan, Beckmann, Kerri, Denton, Brian, Hayen, Andrew, Boutros, Paul, Guo, Wei, Benfante, Nicole, Cowan, Janet, Patil, Dattatraya, Park, Lauren, Ferrante, Stephanie, Mamedov, Alexandre, LaPointe, Vincent, Crump, Trafford, Stavrinides, Vasilis, Kimberly-Duffell, Jenna, Santaolalla, Aida, Olivier, Jonathan, Rancati, Tiziana, Ahlgren, Helén, Mascarós, Juanma, Löfgren, Annica, Lehmann, Kurt, Han Lin, Catherine, Cusick, Thomas, Hirama, Hiromi, Suk Lee, Kwang, Jenster, Guido, Auvinen, Anssi, Haider, Masoom, van Bochove, Kees, Kouspou, Michelle, Paich, Kellie, Helleman, Jozien, Shiong Lee, Lui, and Elhage, Oussama

Published
2021
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18. Efficiency and controllability of stochastic boolean function generation by a random network of non-linear nanoparticle junctions.

Author

Martini, G., Tentori, E., Mirigliano, M., Galli, D. E., Milani, P., Mambretti, F., Moretti, Paolo, and Cusick, Thomas

Subjects
BOOLEAN functions, NANOPARTICLES, BIOLOGICAL neural networks, GOLD films, SEARCH algorithms
Abstract

Amid efforts to address energy consumption in modern computing systems, one promising approach takes advantage of random networks of non-linear nanoscale junctions formed by nanoparticles as substrates for neuromorphic computing. These networks exhibit emergent complexity and collective behaviors akin to biological neural networks, characterized by self-organization, redundancy, and non-linearity. Based on this foundation, a generalization of n-inputs devices has been proposed, where the associated weights depend on all the input values. This model, called receptron, has demonstrated its capability to generate Boolean functions as output, representing a significant breakthrough in unconventional computing methods. In this work, we characterize and present two actual implementations of this paradigm. One approach leverages the nanoscale properties of cluster-assembled Au films, while the other utilizes the recently introduced Stochastic Resistor Network (SRN) model. We first provide a concise overview of the electrical properties of these systems, emphasizing the insights gained from the SRN regarding the physical processes within real nanostructured gold films at a coarse-grained scale. Furthermore, we present evidence indicating the minimum complexity level required by the SRN model to achieve a stochastic dynamics adequate to effectively model a novel component for logic systems. To support our argument that these systems are preferable to conventional random search algorithms, we discuss quantitative criteria based on Information-theoretic tools. This suggests a practical means to steer the stochastic dynamics of the system in a controlled way, thus focusing its random exploration where it is most useful. [ABSTRACT FROM AUTHOR]

Published
2024
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22. Affine equivalence of cubic homogeneous rotation symmetric Boolean functions

Author

Cusick, Thomas W.

Subjects
Computer Science - Information Theory, Mathematics - Number Theory, 94C10, 94A15, 06E30
Abstract

Homogeneous rotation symmetric Boolean functions have been extensively studied in recent years because of their applications in cryptography. Little is known about the basic question of when two such functions are affine equivalent. The simplest case of quadratic rotation symmetric functions which are generated by cyclic permutations of the variables in a single monomial was only settled in 2009. This paper studies the much more complicated cubic case for such functions. A new concept of \emph{patterns} is introduced, by means of which the structure of the smallest group G_n, whose action on the set of all such cubic functions in $n$ variables gives the affine equivalence classes for these functions under permutation of the variables, is determined. We conjecture that the equivalence classes are the same if all nonsingular affine transformations, not just permutations, are allowed. This conjecture is verified if n < 22. Our method gives much more information about the equivalence classes; for example, in this paper we give a complete description of the equivalence classes when n is a prime or a power of 3., Comment: This is the author's version of a manuscript which has been accepted by Information Sciences. The ancillary Mathematica file contains a program which computes the equivalence classes described in the paper, given n = number of variables

Published
2010
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23. Balanced Symmetric Functions over $GF(p)$

Author

Cusick, Thomas W., Li, Yuan, and Stanica, Pantelimon

Subjects
Mathematics - Combinatorics, 05A10, 05A16, 11T71
Abstract

Under mild conditions on $n,p$, we give a lower bound on the number of $n$-variable balanced symmetric polynomials over finite fields $GF(p)$, where $p$ is a prime number. The existence of nonlinear balanced symmetric polynomials is an immediate corollary of this bound. Furthermore, we conjecture that $X(2^t,2^{t+1}l-1)$ are the only nonlinear balanced elementary symmetric polynomials over GF(2), where $X(d,n)=\sum_{i_1

Published
2006

24. Fast Evaluation, Weights and Nonlinearity of Rotation-Symmetric Functions

Author

Cusick, Thomas W. and Stanica, Pantelimon

Subjects
Mathematics - Combinatorics, 05A19, 06E30, 11T71, 94A60, 94C10
Abstract

We study the nonlinearity and the weight of the rotation-symmetric (RotS) functions defined by Pieprzyk and Qu. We give exact results for the nonlinearity and weight of 2-degree RotS functions with the help of the semi-bent functions and we give the generating function for the weight of the 3-degree RotS function. Based on the numerical examples and our observations we state a conjecture on the nonlinearity and weight of the 3-degree RotS functions., Comment: 20 pages; submitted to Discrete Mathematics

Published
2000

26. Outcomes for active surveillance are similar for men with favourable risk ISUP-2 to those with ISUP-1 prostate cancer: A pair matched cohort study

Author

Katelaris, Athos, primary, Amin, Amer, additional, Blazevski, Alexandar, additional, Scheltema, Matthijs J, additional, Cusick, Thomas, additional, Farraha, Melad, additional, Barreto, Daniela, additional, Haynes, Anne Maree, additional, Gondoputro, William, additional, Agrawal, Shikha, additional, Stricker, Phillip, additional, and Thompson, James, additional

Published
2023
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27. Affine equivalence of monomial rotation symmetric Boolean functions: A Pólya’s theorem approach

Author

Cusick Thomas W., Lakshmy K. V., and Sethumadhavan M.

Subjects
affine equivalence, rotation symmetry, boolean functions, pólya’s counting theorem, 94c10, 94a60, 06e30, Mathematics, QA1-939
Abstract

Two Boolean functions are affine equivalent if one can be obtained from the other by applying an affine transformation to the input variables. For a long time, there have been efforts to investigate the affine equivalence of Boolean functions. Due to the complexity of the general problem, only affine equivalence under certain groups of permutations is usually considered. Boolean functions which are invariant under the action of cyclic rotation of the input variables are known as rotation symmetric (RS) Boolean functions. Due to their speed of computation and the prospect of being good cryptographic Boolean functions, this class of Boolean functions has received a lot of attention from cryptographic researchers. In this paper, we study affine equivalence for the simplest rotation symmetric Boolean functions, called MRS functions, which are generated by the cyclic permutations of a single monomial. Using Pólya’s enumeration theorem, we compute the number of equivalence classes, under certain large groups of permutations, for these MRS functions in any number n of variables. If n is prime, we obtain the number of equivalence classes under the group of all permutations of the variables.

Published
2016
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30. Theory of 3-rotation symmetric cubic Boolean functions

Author

Cusick Thomas W. and Cheon Younhwan

Subjects
boolean functions, rotation symmetry, affine equivalence, equivalence class, hamming weight, recursions, 94c10, 94a15, 06e30, Mathematics, QA1-939
Abstract

Rotation symmetric Boolean functions have been extensively studied in the last 15 years or so because of their importance in cryptography and coding theory. Until recently, very little was known about such basic questions as when two such functions are affine equivalent. This question in important in applications, because almost all important properties of Boolean functions (such as Hamming weight, nonlinearity, etc.) are affine invariants, so when searching a set for functions with useful properties, it suffices to consider just one function in each equivalence class. This can greatly reduce computation time. Even for quadratic functions, the analysis of affine equivalence was only completed in 2009. The much more complicated case of cubic functions was completed in the special case of affine equivalence under permutations for monomial rotation symmetric functions in two papers from 2011 and 2014. There has also been recent progress for some special cases for functions of degree >3${> 3}$. In 2007 it was found that functions satisfying a new notion of k-rotation symmetry for k > 1 (where the case k = 1 is ordinary rotation symmetry) were of substantial interest in cryptography and coding theory. Since then several researchers have used these functions for k = 2 and 3 to study such topics as construction of bent functions, nonlinearity and covering radii of various codes. In this paper we develop a detailed theory for the monomial 3-rotation symmetric cubic functions, extending earlier work for the case k = 2 of these functions.

Published
2015
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31. Event‐free survival after radical prostatectomy according to prostate‐specific membrane antigen‐positron emission tomography and European Association of Urology biochemical recurrence risk groups

Author

Roberts, Matthew J., primary, Chatfield, Mark D., additional, Hruby, George, additional, Nandurkar, Rohan, additional, Roach, Paul, additional, Watts, Jo Anne, additional, Cusick, Thomas, additional, Kneebone, Andrew, additional, Eade, Thomas, additional, Ho, Bao, additional, Nguyen, Andrew, additional, Tang, Colin, additional, McCarthy, Michael, additional, Francis, Roslyn, additional, Stricker, Phillip, additional, and Emmett, Louise, additional

Published
2022
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41. Preface

Author

Cusick, Thomas W., primary and Stănică, Pantelimon, additional

Published
2017
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45. Localisation and expression of aquaporin-4 around arterioles, venules and capillaries in the rat spinal cord

Author

Cusick, Thomas

Subjects
sense organs, Other education not elsewhere classified
Abstract

The traditional view of cerebrospinal fluid (CSF) circulation within the central nervous system (CNS) is that CSF is produced within the choroid plexus, flows slowly through the subarachnoid space (SAS). and is reabsorbed by arachnoid villi or around spinal nerves. Despite acceptance of this viewpoint, the mechanism by which CSF is moved around the CNS is not fully understood. Our current understanding of CSF flow within the brain is that fluid inflow occurs predominantly around peri-arterial spaces, whilst outflow occurs around peri-venular spaces. This fluid movement is believed to be mediated by the protein aquaporin-4 (AQP4). Whether AQP4 contributes to the movement of CSF within the spinal cord remains to be determined. The aim of the following study was to assess AQP4 expression in the rodent spinal cord in relation to microvascular structures. Multiple labelling immunohistochemistry was performed on 6 and 12 week old rats to distinguish arterioles from venules, and capillaries were identified by measuring the diameter of each stained vessel. AQP4 was found around both arterioles and venules within the rodent spinal cord, and was expressed in higher levels in the younger rats. No overall difference was found between AQP4 expression levels around arterioles, venules and capillaries, however, expression of AQP4 was found to decrease as the diameter of arterioles increased. The results of this study lay the foundation for future experiments to assess the effect of AQP4 antagonism on fluid flow within the spinal cord.

Published
2022
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49. Recursive weights for some Boolean functions

Author

Brown Alyssa and Cusick Thomas W.

Subjects
boolean functions, rotation symmetry, hamming weight, recursion, Mathematics, QA1-939
Abstract

This paper studies degree 3 Boolean functions in n variables which are rotation symmetric, that is, invariant under any cyclic shift of the indices of the variables. These rotation symmetric functions have been extensively studied in the last dozen years or so because of their importance in cryptography. Let denote the (Hamming) weight of the Boolean function f. We extend and simplify results of Bileschi, Cusick and Padgett, who gave an algorithm for finding a recursion for the sequence , where is the cubic rotation symmetric function generated by the monomial We show that the weights for the function (note this is just the sum of the first terms in the definition of ) satisfy the same recursion as the weights for . We prove the recursion for the weights of by a method based on the work by Bileschi–Cusick–Padgett, but we are able to avoid a lot of the complications in their proofs. In particular, we do not need the technical notion of “g terms” or the elaborate proof of similarity of certain matrices.

Published
2012
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50. GUIDE: a randomised non-comparative phase II trial of biomarker-driven intermittent docetaxel versus standard-of-care docetaxel in metastatic castration-resistant prostate cancer (clinical trial protocol)

Author

Conduit, Ciara, primary, Mak, Blossom, additional, Qu, Wenjia, additional, Lulio, Juliana Di, additional, Burder, Ronan, additional, Bressel, Matthias, additional, Cusick, Thomas, additional, Dhillon, Haryana M., additional, Lourenço, Richard De Abreu, additional, Underhill, Craig, additional, Torres, Javier, additional, Crumbaker, Megan, additional, Honeyball, Florian, additional, Linton, Anthony, additional, Allen, Ray, additional, Davis, Ian D., additional, Clark, Susan J., additional, Horvath, Lisa G., additional, and Mahon, Kate L., additional

Published
2022
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