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For a delay difference equation , gk(n) ≤ n, K > 0, a connection between oscillation properties of this equation and the corresponding linear equations is established. Explicit nonoscillation and oscillation conditions are presented. Positiveness of solutions is discussed.

We explore stabilization for nonlinear systems of difference equations with modified Target-Oriented Control and a chosen equilibrium as a target, both in deterministic and stochastic settings. The influence of stochastic components in the control parameters is explored. The results are tested on the Hénon and the Lozi maps. [ABSTRACT FROM AUTHOR]

Abdalla Bowirrat,1 Igor Elman,2 Catherine A Dennen,3 Marjorie C Gondré-Lewis,4 Jean Lud Cadet,5 Jag Khalsa,6 David Baron,7 Diwanshu Soni,8 Mark S Gold,9 Thomas J McLaughlin,10 Debasis Bagchi,11 Eric R Braverman,12 Mauro Ceccanti,13 Panayotis K Thanos,14,15 Edward Justin Modestino,16 Keerthy Sunder,17,18 Nicole Jafari,19,20 Foojan Zeine,21,22 Rajendra D Badgaiyan,23 Debmalya Barh,24,25 Milan Makale,26 Kevin T Murphy,27 Kenneth Blum1,7,10,12,18,20,24,28,29 1Department of Molecular Biology, Adelson School of Medicine, Ariel University, Ariel, 40700, Israel; 2Cambridge Health Alliance, Harvard Medical School, Cambridge, MA, 02139, USA; 3Department of Family Medicine, Jefferson Health Northeast, Philadelphia, PA, USA; 4Neuropsychopharmacology Laboratory, Department of Anatomy, Howard University College of Medicine, Washington, DC, 20059, USA; 5Molecular Neuropsychiatry Research Branch, NIH National Institute on Drug Abuse, Bethesda, MD, 20892, USA; 6Department of Microbiology, Immunology and Tropical Medicine, George Washington University, School of Medicine, Washington, DC, USA; 7Division of Addiction Research & Education, Center for Sports, Exercise, and Mental Health, Western University of Health Sciences, Pomona, CA, 91766, USA; 8Western University Health Sciences School of Medicine, Pomona, CA, USA; 9Department of Psychiatry, Washington University School of Medicine, St. Louis, MO, 63110, USA; 10Division of Reward Deficiency Clinics, TranspliceGen Therapeutics, Inc, Austin, TX, USA; 11Department of Pharmaceutical Sciences, Texas Southern University College of Pharmacy, Houston, TX, USA; 12Division of Clinical Neurology, The Kenneth Blum Institute of Neurogenetics & Behavior, LLC, Austin, TX, USA; 13Alcohol Addiction Program, Latium Region Referral Center, Sapienza University of Rome, Roma, 00185, Italy; 14Behavioral Neuropharmacology and Neuroimaging Laboratory on Addictions, Clinical Research Institute on Addictions, Department of Pharmacology and Toxicology, Jacobs School of Medicine and Biosciences, State University of New York at Buffalo, Buffalo, NY, 14203, USA; 15Department of Psychology, State University of New York at Buffalo, Buffalo, NY, 14203, USA; 16Department of Psychology, Curry College, Milton, MA, USA; 17Karma Doctors & Karma TMS, and Suder Foundation, Palm Springs, CA, USA; 18Department of Medicine, University of California, Riverside School of Medicine, Riverside, CA, USA; 19Department of Human Development, California State University at Long Beach, Long Beach, CA, USA; 20Division of Personalized Medicine, Cross-Cultural Research and Educational Institute, San Clemente, CA, USA; 21Awareness Integration Institute, San Clemente, CA, USA; 22Department of Health Science, California State University at Long Beach, Long Beach, CA, USA; 23Department of Psychiatry, Mt. Sinai School of Medicine, New York, NY, USA; 24Centre for Genomics and Applied Gene Technology, Institute of Integrative Omics and Applied Biotechnology (IIOAB), Purba Medinipur, WB, 721172, India; 25Departamento de Genética, Ecologia e Evolução, Instituto de Ciências Biológicas, Universidade Federal de Minas Gerais, Belo Horizonte, 31270-901, Brazil; 26Department of Radiation Medicine and Applied Sciences, UC San Diego, La Jolla, CA, 92093-0819, USA; 27Department of Radiation Oncology, University of California San Diego, La Jolla, CA, USA; 28Department of Psychiatry, University of Vermont School of Medicine, Burlington, VA, USA; 29Institute of Psychology, ELTE Eötvös Loránd University, Budapest, HungaryCorrespondence: Igor Elman; Kenneth Blum, Email dr.igorelman@gmail.com; drd2gene@gmail.comAbstract: Loneliness, an established risk factor for both, mental and physical morbidity, is a mounting public health concern. However, the neurobiological mechanisms underlying loneliness-related morbidity are not yet well defined. Here we examined the role of genes and associated DNA risk polymorphic variants that are implicated in loneliness via genetic and epigenetic mechanisms and may thus point to specific therapeutic targets. Searches were conducted on PubMed, Medline, and EMBASE databases using specific Medical Subject Headings terms such as loneliness and genes, neuro- and epigenetics, addiction, affective disorders, alcohol, anti-reward, anxiety, depression, dopamine, cancer, cardiovascular, cognitive, hypodopaminergia, medical, motivation, (neuro)psychopathology, social isolation, and reward deficiency. The narrative literature review yielded recursive collections of scientific and clinical evidence, which were subsequently condensed and summarized in the following key areas: (1) Genetic Antecedents: Exploration of multiple genes mediating reward, stress, immunity and other important vital functions; (2) Genes and Mental Health: Examination of genes linked to personality traits and mental illnesses providing insights into the intricate network of interaction converging on the experience of loneliness; (3) Epigenetic Effects: Inquiry into instances of loneliness and social isolation that are driven by epigenetic methylations associated with negative childhood experiences; and (4) Neural Correlates: Analysis of loneliness-related affective states and cognitions with a focus on hypodopaminergic reward deficiency arising in the context of early life stress, eg, maternal separation, underscoring the importance of parental support early in life. Identification of the individual contributions by various (epi)genetic factors presents opportunities for the creation of innovative preventive, diagnostic, and therapeutic approaches for individuals who cope with persistent feelings of loneliness. The clinical facets and therapeutic prospects associated with the current understanding of loneliness, are discussed emphasizing the relevance of genes and DNA risk polymorphic variants in the context of loneliness-related morbidity.Keywords: addiction, affective, alcohol, anti-reward, anxiety, depression, dopamine, cancer, genes, cardiovascular, cognitive, medical, motivation, social isolation, reward deficiency