Dyadic Expansion Research Articles

Hausdorff dimension of certain sets arising by the maximal run-length function over factorial language

Let A be a language with infinite number of words over the alphabet Σ={0,1} and let (xi)i=1+∞ be the digits sequence in the unique terminating dyadic expansion of x∈[0,1). The maximal run-length function ln(x,A) of x over A is defined byln(x,A)=max⁡{k:xi+1xi+2⋯xi+k∈Afor some0≤i≤n−k}. In a general setting, Wu (2024) [18] proved a metrical property on the asymptotic behavior of ln(x,A), which extends the famous Erdős-Rényi limit theorem that represents for the special case A={1k:k≥1}. Under the assumption that A is factorial (i.e. the language A is closed under subword), Wu studied the exceptional setE={x∈[0,1):limn→∞⁡ln(x,A)mn=0andlimn→∞⁡ln(x,A)=∞}, and he obtained a lower bound of the Hausdorff dimension of E with some conditions on mn. In this paper, we determine the Hausdorff dimension of E without any additional assumption on mn and we obtain the Hausdorff dimension of two other kinds of exceptional sets of elements whose maximal run-length functions have different growth rates. Our results significantly extend the existing dimension results in this topic.

Tronick's still face experiments: their relevance to couple therapy

The author describes the work of Tronick and associated mother–infant researchers, who have added to a deep understanding of analyst–patient dyadic interactions. As an analytically oriented couples therapist, the author is particularly interested in applying Tronick's dyadic system to the triadic one that is couples therapy. A clinical example is used to explore Tronick's concepts of interactive regulation, mismatch, rupture and repair, meaning making, dyadic expansion of consciousness, and the here and now, as they relate to couples therapy.

The Moore–Penrose Inverse and Singular Value Decomposition of Split Quaternions

Using the concept of a transposition anti-involution in Clifford algebra $$C \, \ell _{1,1}$$ and the isomorphisms $${\mathbb {H}}_s \cong C \, \ell _{1,1} \cong \text {Mat}(2,{\mathbb {R}}),$$ where $${\mathbb {H}}_s$$ is the algebra of split quaternions and $$\text {Mat}(2,{\mathbb {R}})$$ is the algebra of $$2 \times 2$$ real matrices, one can find the Moore–Penrose inverse $$q^{+}$$ of a non-zero non-invertible split quaternion q. In particular, using a well-known algorithm for finding the Moore-Penrose inverse $$Q^{+}$$ of a non-zero $$2 \times 2$$ matrix Q of rank 1, one can give four governing equations that the (unique) split quaternion $$q^{+}$$ corresponding to $$Q^{+}$$ must satisfy. We show how a dyadic expansion and a Singular Value Decomposition can be found for any split quaternion q, and we relate them to $$q^{+}$$ . Results presented in this paper may be useful in a plethora of recent applications of split quaternions.

Psychotherapy beyond all the words: Dyadic expansion, vagal regulation, and biofeedback in psychotherapy.

Psychotherapy beyond all the words: Dyadic expansion, vagal regulation, and biofeedback in psychotherapy.

On the range of simple symmetric random walks on the line

This paper is aimed at a detailed study of the behaviors of random walks which is defined by the dyadic expansions of points. More precisely, let x=(ϵ1(x),ϵ2(x),…) be the dyadic expansion for a point x∈[0,1) and Sn(x)=∑k=1n(2ϵk(x)−1), which can be regarded as a simple symmetric random walk on Z. Denote by Rn(x) the cardinality of the set {S1(x),…,Sn(x)}, which is just the distinct position of x passed after n times. The set of points whose behavior satisfies Rn(x)∼cnγ is studied (c>0 and 0<γ≤1 being fixed) and its Hausdorff dimension is calculated.

EXCEPTIONAL SETS RELATED TO THE RUN-LENGTH FUNCTION OF BETA-EXPANSIONS

Let [Formula: see text] and [Formula: see text] be real numbers. The run-length function of [Formula: see text]-expansions denoted by [Formula: see text] is defined as the maximal length of consecutive zeros in the first [Formula: see text] digits of the [Formula: see text]-expansion of [Formula: see text]. It is known that for Lebesgue almost all [Formula: see text], [Formula: see text] increases to infinity with the logarithmic speed [Formula: see text] as [Formula: see text] goes to infinity. In this paper, we calculate the Hausdorff dimension of the subtle set for which [Formula: see text] grows to infinity with other speeds. More precisely, we prove that for any [Formula: see text], the set [Formula: see text] has full Hausdorff dimension, where [Formula: see text] is a strictly increasing function satisfying that [Formula: see text] is non-increasing, [Formula: see text] and [Formula: see text] as [Formula: see text]. This result significantly extends the existing results in this topic, such as the results in [J.-H. Ma, S.-Y. Wen and Z.-Y. Wen, Egoroff’s theorem and maximal run length, Monatsh. Math. 151(4) (2007) 287–292; R.-B. Zou, Hausdorff dimension of the maximal run-length in dyadic expansion, Czechoslovak Math. J. 61(4) (2011) 881–888; J.-J. Li and M. Wu, On exceptional sets in Erdős–Rényi limit theorem, J. Math. Anal. Appl. 436(1) (2016) 355–365; J.-J. Li and M. Wu, On exceptional sets in Erdős–Rényi limit theorem revisited, Monatsh. Math. 182(4) (2017) 865–875; Y. Sun and J. Xu, A remark on exceptional sets in Erdős–Rényi limit theorem, Monatsh. Math. 184(2) (2017) 291–296; X. Tong, Y.-L. Yu and Y.-F. Zhao, On the maximal length of consecutive zero digits of [Formula: see text]-expansions, Int. J. Number Theory 12(3) (2016) 625–633; J. Liu, and M.-Y. Lü, Hausdorff dimension of some sets arising by the run-length function of [Formula: see text]-expansions, J. Math. Anal. Appl. 455(1) (2017) 832–841; L.-X. Zheng, M. Wu and B. Li, The exceptional sets on the run-length function of [Formula: see text]-expansions, Fractals 25(6) (2017) 1750060; X. Gao, H. Hu and Z.-H. Li, A result on the maximal length of consecutive 0 digits in [Formula: see text]-expansions, Turkish J. Math. 42(2) (2018) 656–665, doi: 10.3906/mat-1704-119].

Blind Hyperspectral Unmixing Using Total Variation and &lt;inline-formula&gt; &lt;tex-math notation="LaTeX"&gt;$\ell_q$&lt;/tex-math&gt; &lt;/inline-formula&gt; Sparse Regularization

Blind hyperspectral unmixing involves jointly estimating endmembers and fractional abundances in hyperspectral images. An endmember is the spectral signature of a specific material in an image, while an abundance map specifies the amount of a material seen in each pixel in an image. In this paper, a new cyclic descent algorithm for blind hyperspectral unmixing using total variation (TV) and $\boldsymbol{\ell_q}$ sparse regularization is proposed. Abundance maps are both spatially smooth and sparse. Their sparsity derives from the fact that each material in the image is not represented in all pixels. The abundance maps are assumed to be piecewise smooth since adjacent pixels in natural images tend to be composed of similar material. The TV regularizer is used to encourage piecewise smooth images, and the $\boldsymbol{\ell_q}$ regularizer promotes sparsity. The dyadic expansion decouples the problem, making a cyclic descent procedure possible, where one abundance map is estimated, followed by the estimation of one endmember. A novel debiasing technique is also employed to reduce the bias of the algorithm. The algorithm is evaluated using both simulated and real hyperspectral images.

On exceptional sets in Erdős–Rényi limit theorem revisited

For $$x\in [0,1],$$ the run-length function $$r_n(x)$$ is defined as the length of the longest run of 1’s amongst the first n dyadic digits in the dyadic expansion of x. Let H denote the set of monotonically increasing functions $$\varphi :\mathbb {N}\rightarrow (0,+\infty )$$ with $$\lim _{n\rightarrow \infty }\varphi (n)=+\infty $$ . For any $$\varphi \in H$$ , we prove that the set $$\begin{aligned} E_{\max }^\varphi =\left\{ x\in [0,1]:\liminf \limits _{n\rightarrow \infty }\frac{r_n(x)}{\varphi (n)}=0, \limsup \limits _{n\rightarrow \infty }\frac{r_n(x)}{\varphi (n)}=+\infty \right\} \end{aligned}$$ either has Hausdorff dimension one and is residual in [0, 1] or is empty. The result solves a conjecture posed in Li and Wu (J Math Anal Appl 436:355–365, 2016) affirmatively.

Dyadically Expanded States of Consciousness and Therapeutic Change in the Interaction Between Analyst and Adult Patient

This paper presents a discussion of the mutual regulation model and the dyadic expansion of consciousness model as a way to better understand the interaction between the analysand and analyst and the process of therapeutic change. We argue that by examining matching, mismatching, and in particular reparation of therapeutic exchanges at the micro-level as well as the macro-level of the therapeutic exchange, including whole sessions or even over multiple sessions provides insight in the establishment of the therapeutic relationship and the formation of Dyadic States of Consciousness. Clinical examples are presented.

On exceptional sets in Erdős–Rényi limit theorem

For x∈[0,1], the run-length function rn(x) is defined as the length of the longest run of 1's amongst the first n dyadic digits in the dyadic expansion of x. Erdős and Rényi proved that limn→∞⁡rn(x)log2⁡n=1 for Lebesgue almost all x∈[0,1]. In this paper, we study the Hausdorff dimensions of the exceptional sets in Erdős–Rényi limit theorem. Let φ:N→(0,+∞) be a monotonically increasing function satisfying limn→∞⁡nφ(n1+α)=+∞ with some 0<α≤1. We prove that the setEmaxφ={x∈[0,1]:liminfn→∞rn(x)φ(n)=0,limsupn→∞rn(x)φ(n)=+∞} has Hausdorff dimension one and is residual in [0,1].

Therapeutic Creativity, Complexity, and Ed Tronick’s Dyadic Expansion of Consciousness

Ed Tronick’s research on interaction and infant development has greatly inspired clinicians. His new focus on the infant resulted in changes in ways of thinking about infant development and in the psychotherapeutic methods used with very young infants and their caregivers. The mutual regulation model (MRM) gave empirical support to clinical observations and later gave rise to the Still Face paradigm. The MRM model offers a theoretical framework for developing detailed mother–infant therapeutic capacities while it raises major theoretical questions on the manners in which the infant’s personal inner world develops. Considering the parent–infant dyad as a whole underscores the need to address infant and parental characteristics together, integrating development itself as an important force for mental health. This suggests that the infant is both a system in itself, capable of self-organization, and a part of a cocreating system with his or her partner, for example in the dyadic organization of states. It could mean that analysts should probably be both modest and brazen in their therapeutic interventions, taking advantage, as clinicians of knowledge based on the latest scientific research, linked to experience and culturally appropriate socially sensitive interventions. In this way, they may provide positively enriched experiences, working through problematic negative emotions that our patients will keep and use in their own creative way.

Even More Than the “Something More”: Tronick’s Dyadic Expansion of Consciousness Model and the Expansion of Child Psychoanalytic Technique

Contemporary psychoanalysis considers itself to be a discipline fundamentally concerned with meaning and meaning-making processes. Ed Tronick’s research provides scientific support for the theoretical position that meaning making is a central process in psychological development and in mental health/illness. His work collaborating with psychoanalysts has made major contributions to the psychoanalytic literature on therapeutic action, with a special emphasis on the means by which implicit meanings are activated and modified in analytic treatment—the something more than interpretation. This article is about a different something more, the even more that psychoanalytic theory and technique can evolve through further incorporating Tronick’s important findings. Tronick’s Dyadic Expansion of Consciousness Model will be briefly reviewed—emphasizing his conceptualization of meanings as being composed of multiple commingling layers (biological, psychological, relational, and social) coming together in a nonlinear “messy” mixture of mutually influencing (both bottom-up and top-down) currents. This multilayered model of meaning opens up the reconsideration of an exciting array of technical options—traditionally considered nonanalytic—to be reunderstood as truly psychoanalytic in that they address one or more of the implicit or explicit levels of meaning that a patient makes of his or her self and world. Examples of these interventions include parent work, work with teachers and schools, as well as interventions adapted from other disciplines such as Occupational Therapy. These technical possibilities are illustrated using case material from the psychoanalytic treatment of a nine-year-old boy.

On highly regular embeddings

A continuous map $\mathbb {R}^d\to \mathbb {R}^N$ is $k$-regular if it maps any $k$ pairwise distinct points to $k$ linearly independent vectors. Our main result on $k$-regular maps is the following lower bound for the existence of such maps between Euclidean spaces, in which $\alpha (k)$ denotes the number of ones in the dyadic expansion of $k$: For $d\geq 1$ and $k\geq 1$ there is no $k$-regular map $\mathbb {R}^d\to \mathbb {R}^N$ for $N<d(k-\alpha (k))+\alpha (k)$. This reproduces a result of Cohen & Handel from 1978 for $d=2$ and the extension by Chisholm from 1979 to the case when $d$ is a power of $2$; for the other values of $d$ our bounds are in general better than Karasev’s (2010), who had only recently gone beyond Chisholm’s special case. In particular, our lower bound turns out to be tight for $k\le 3$. A framework of Cohen & Handel (1979) relates the existence of a $k$-regular map to the existence of a low-dimensional inverse of a certain vector bundle. Thus the non-existence of regular maps into $\mathbb {R}^N$ for small $N$ follows from the non-vanishing of specific dual Stiefel–Whitney classes. This we prove using the general Borsuk–Ulam–Bourgin–Yang theorem combined with a key observation by Hung (1990) about the cohomology algebras of configuration spaces. Our study produces similar lower bound results also for the existence of $\ell$-skew embeddings $\mathbb {R}^d\to \mathbb {R}^N$, for which we require that the images of the tangent spaces of any $\ell$ distinct points are skew affine subspaces. This extends work by Ghomi & Tabachnikov (2008) for the case $\ell =2$.

SINGULARITY ORDER OF THE RIESZ-NÁGY-TAKÁCS FUNCTION

Abstract. We give the characterization of H¨older differentiability pointsand non-differentiability points of the Riesz-N´agy-Taka´cs (RNT) singularfunction Ψ a,p satisfying Ψ a,p (a) = p. It generalizes recent multifractaland metric number theoretical results associated with the RNT function.Besides, we classify the singular functions using the singularity order de-duced from the H¨older derivative giving the information that a strictlyincreasing smooth function having a positive derivative Lebesgue almosteverywhere has the singularity order 1 and the RNT function Ψ a,p hasthe singularity order g(a,p) = alogp+(1−a)log(1−p)aloga+(1−a)log(1−a) ≥1. 1. IntroductionRecently many ([8, 9, 10, 16]) studied the Cantor function, a singular func-tion which is not strictly increasing and the Minkowski’s Question Mark func-tion which is a strictly increasing singular function. Also J. Parad´isetal. ([17])studied some conditions of the null and infinite derivatives of the RNT strictlyincreasing singular function using metric number theory.Recently we ([4]) also studied multifractal characterization of the null andinfinite derivative sets and the non-differentiability set of the RNT singularfunction, which is the typical singular function related to mutifractal theory.In this paper, we employ the Ho¨lder derivative, which is a generalized form ofthe usual derivative, of the RNT function on the unit interval. This definitionextends the concept of the singularity to the general singularity for a strictlyincreasing continuous function. For every point in the unit interval, we ([4])can give some code or dyadic expansion using digit 0 and 1, generating thedistribution set determined by the frequency of the zero in its expansion. Thedistribution sets in the unit interval are the local dimension sets by a self-similar

Hausdorff dimension of the maximal run-length in dyadic expansion

For any x ∈ [0, 1), let x = [ɛ 1, ɛ 2, …,] be its dyadic expansion. Call r n (x):= max{j ⩾ 1: ɛ i+1 = … = ɛ i+j = 1, 0 ⩽ i ⩽ n − j} the n-th maximal run-length function of x. P.Erdos and A.Renyi showed that $$\mathop {\lim }\limits_{n \to \infty } $$ r n (x)/log2 n = 1 almost surely. This paper is concentrated on the points violating the above law. The size of sets of points, whose runlength function assumes on other possible asymptotic behaviors than log2 n, is quantified by their Hausdorff dimension.

“The Noise Monitor”

This paper discusses a contribution of developmental theory to the psychoanalytic concept of "the talking cure." The developmental theory presented is the dyadic expansion of consciousness model (Tronick 2007), a model consistent with the principles of nonlinear systems theory. The concept of "polysemic bundles" as a way of understanding the multiple simultaneous meaning-making processes occurring in dyadic communication is introduced. The theoretical discussion--focused primarily on the analysis of children--is illustrated with descriptions of videotaped sequences from the first session in the analysis of a five-year-old boy. The relevance of these insights to the analytic treatment of adults is then considered.

The transcendent function, moments of meeting and dyadic consciousness: constructive and destructive co‐creation in the analytic dyad

In reading the work of Beebe (2002), Sander (Amadei & Bianchi 2008), Tronick (2007) and Stern and the Boston Change Process Study Group (1998), resonances to the transcendent function can be registered but these researchers seem to be more focused on the interpersonal domain. In particular Tronick's concept of 'dyadic expansion of consciousness' and 'moments of meeting' from the Boston Change Process Study Group describe external dyadic interactions between mothers and babies and therapists and patients while, in contrast, Jung's early focus was on the intrapsychic process of internal interaction between conscious and unconscious within an individual. From an overall perspective, the interpersonal process of change described by infant researchers, when held in conjunction with Jung's internal process of change, together form a transcendent whole that could also be called a complex adaptive system. Such new theoretical perspectives from other fields confirm and elaborate long held Jungian notions such as the transcendent function which is, in many ways, harmonious with a systems perspective. Throughout this paper, clinical vignettes of interactive moments along with sand play and dreams will be used to illustrate theoretical points regarding the healthy process of the transcendent function along with descriptions of failures of such conjunctive experiences.

Dimensions of weighted Besicovitch sets

We consider weighted Besicovitch sets which are defined in terms of the weighted frequency of digit 1 in the dyadic expansion of real numbers. Explicit formulas for their Hausdorff dimensions are given.

Pattern Occurrence in the Dyadic Expansion of Square Root of Two and an Analysis of Pseudorandom Number Generators

AbstractRecently, designs of pseudorandom number generators (PRNGs) using integer-valued variants of logistic maps and their applications to certain cryptographic schemes have been studied, due mostly to their ease of implementation and performance. However, it has been noted that this ease is reduced for some choices of the PRNGs accuracy parameters. In this article, we show that the distribution of such undesirable accuracy parameters is closely related to the occurrence of some patterns in the dyadic expansion of the square root of 2. We prove that for an arbitrary infinite binary word, the asymptotic occurrence rate of these patterns is bounded in terms of the asymptotic occurrence rate of zeroes. As a consequence, a classical conjecture on asymptotic evenness of occurrence of zeroes and ones in the dyadic expansion of the square root of 2 implies that the asymptotic rate of the undesirable accuracy parameters for the PRNGs is at least 1/6.

Setting Up the Doll House: A Developmental Perspective on Termination

This article considers the contribution that consideration of the moment-to-moment process in psychoanalysis can make to an understanding of termination. Information on moment-to-moment interactions related to termination is developed from videotape microanalysis of termination discussions—focusing on a child analysis, but including an example from an adult analysis—as viewed through the structure of a developmental model (Tronick's dyadic expansion of consciousness model) that is consistent with dynamic systems theory. The theory emphasizes the co-creation of meaning that occurs in an analysis, as indicated by verbal, as well as nonverbal, interactions that are apparent in the videotape. The insights from this approach can enrich the traditional psychoanalytic views on termination by emphasizing the ongoing developmental process, with termination more of an important step in this process than a specific ending.