A generalization of cyclic difference sets I

AbstractThis is the first of two papers on addition sets. In this paper, the basic properties of addition sets are given. It also contains examples of addition sets arising from natural central groupoids, (0, 1)-matrices satisfying the equation M2 = dI+λJ and Nth power residues. Their relationship with difference sets is also explained

A generalization of cyclic difference sets II

This is the second paper on addition sets. A generalization of Hall's Multiplier Theorem for difference sets is given. Some nonexistence theorems are also given. These methods are used to compile a table of addition sets with parameter k≤10. One unsolved case still remains

Classification of generalized Hadamard matrices H(6,3) and quaternary Hermitian self-dual codes of length 18

All generalized Hadamard matrices of order 18 over a group of order 3, H(6,3), are enumerated in two different ways: once, as class regular symmetric (6,3)-nets, or symmetric transversal designs on 54 points and 54 blocks with a group of order 3 acting semi-regularly on points and blocks, and secondly, as collections of full weight vectors in quaternary Hermitian self-dual codes of length 18. The second enumeration is based on the classification of Hermitian self-dual [18,9] codes over GF(4), completed in this paper. It is shown that up to monomial equivalence, there are 85 generalized Hadamard matrices H(6,3), and 245 inequivalent Hermitian self-dual codes of length 18 over GF(4).Comment: 17 pages. Minor revisio

Rational G-Circulants Satisfying the Matrix Equation A² = dI + λJ

A g-circulant is a square matrix of rational numbers in which each row is obtained from the preceding row by shifting the elements cyclically g columns to the right. This work studies g-circulants A which satisfy the matrix equation A2 = dI + λJ, where I is the identity matrix and J is the matrix of 1's. Necessary and sufficient conditions are given for the existence of solutions when g = 1. The existence of (0,1) g-circulants satisfying A2 = dI + λJ is shown to be equivalent to the existence of (v, k, λ, g)-addition sets, which are generalizations of difference sets. It is proved that there are no nontrivial (v, k, λ, 1)-addition sets. Some examples of (v, k, λ, g)-addition sets are given and the multiplier theorem for (v, k, λ, g)-addition sets is also proved.</p

Mechanochromic, Structurally Colored, and Edible Hydrogels Prepared from Hydroxypropyl Cellulose and Gelatin

Abstract: Hydroxypropyl cellulose (HPC) is an edible, cost‐effective and widely used derivative of cellulose. Under lyotropic conditions in water, HPC forms a photonic, liquid crystalline mesophase with an exceptional mechanochromic response. However, due to insufficient physical cross‐linking photonic HPC can flow freely as a viscous liquid, preventing the exploitation of this mechanochromic material in the absence of any external encapsulation or structural confinement. Here this challenge is addressed by mixing HPC and gelatin in water to form a self‐supporting, viscoelastic, and edible supramolecular photonic hydrogel. It is demonstrated that the structural coloration, mechanochromism and non‐Newtonian shear‐thinning behavior of the lyotropic HPC solutions can all be retained into the gel state. Moreover, the rigidity of the HPC‐gel provides a 69% shorter mechanochromic relaxation time back to its initial color when compared to the liquid HPC–water only system, broadening the dynamic color range of HPC by approximately 2.5× in response to a compressive pressure. Finally, the ability to formulate the HPC‐gels in a scalable fashion from only water and “food‐grade” constituents unlocks a wide range of potential applications, from response‑tunable mechanochromic materials and colorant‐free food decoration, to short‐term sensors in, for example, biodegradable “smart labels” for food packaging

Effect of computerized cognitive training on mood, cognition, and serum brain-derived neurotrophic factor level in late-life depression — a pilot randomized controlled trial

BackgroundThe aim of this pilot randomized controlled trial was to test the feasibility of a computerized cognitive training targeting executive dysfunction in late-life depression and to investigate its impact on mood, cognition, and brain-derived neurotrophic factor (BDNF) levels.MethodsA total of 28 community-living Chinese individuals aged 55–75 with moderate-to-severe depression and cognitive symptoms (but without mild cognitive impairment or dementia) were recruited from a community centre in Hong Kong. Participants were randomly allocated to either the experimental (receiving computerized cognitive training) or the control group (receiving computer-based health education). Both programs lasted for one hour and were conducted twice a week for 6 weeks at the community centre. We assessed mood using the Hamilton Rating Scale for Depression (HAM-D) and Patient Health Questionaire-9 (PHQ-9), cognition using the Montreal Cognitive Assessment (MoCA), and serum BDNF levels at baseline and follow-up. We performed repeated measures analysis of variance to compare the differences in outcome changes between groups and correlation analysis to test if changes in mood and cognition correlated with changes in BDNF level.ResultsOur sample had a mean age of 66.8 (SD = 5.3) years, a mean HAM-D score of 19.4 (SD = 7.5), and a mean PHQ-9 score of 18.0 (SD = 6.3). No adverse effects were reported. Significant differences were observed between the experimental and control groups in changes in HAM-D (-8.4 vs. -2.9; group difference = -5.5; p = 0.01), PHQ-9 (-6.6 vs. -0.6; -6.0; p &lt; 0.001), MoCA (1.4 vs. -1.3; 2.7; p = 0.001), and serum BDNF levels (in pg/ml; 2088.3 vs. -3277.4; 5365.6; p = 0.02). Additionally, changes in HAM-D, PHQ-9, and MoCA scores correlated significantly with changes in BDNF level.ConclusionWith computerized cognitive training improving mood and cognition and increasing serum BDNF levels in 6 weeks, it may serve as a safe and effective evidence-based alternative or adjuvant treatment for late-life depression.Clinical trial registrationhttps://www.chictr.org.cn/indexEN.html, identifier ChiCTR1900027029