To Örebro University

An algorithm for solving a class of linear variable order fractional partial differential equations (FPDEs) is presented in this paper. We utilized a combination of Caputo fractional derivatives with the Haar wavelet collocation method (HWCM) and Eiler–Fouirier method to solve linear variable order FPDEs. Fractional calculus is a more complex form of standard mathematical calculus. FPDEs is an equation that uses a fractional derivative (FD) by t > 0 (time variable) and two derivatives by x ∈ R (speciousness variable). FPDEs are used to model a wide range of phenomena in the fields of science, electric circuits, engineering and mechanical systems. In this paper we study solvability of the Cauchy and boundary value problem is constructed for FPDE. Parabolic pseudodifferential equations this pseudodifferential operator with noon-smooth symbols introduced by Eidel’man S. and Drin’ Ya. for the first time. The Haar wavelets, which are made up of piecewise constant functions, are the mathematically simplest of all the wavelet families. These wavelets have the advantage of being analytically integrated any number of times. In Haar wavelet two functions, sclaling and mother function, play an essential role. We may therefore draw the conclusion that the Haar wavelet collocation method and Eiler–Fouirier method is an effective method for finding the solution of linear VO FPDEs.

Objective: Childhood traumatic events are key risk factors for posttraumatic stress disorder (PTSD) and other psychiatric conditions. The war context of the Russian invasion of Ukraine has exposed Ukrainian children to direct and indirect war-related traumatic events and has significantly increased the mental health burden of Ukrainian youth. The objective of the present article is to report the prevalence and correlates of significant PTSD symptoms among children in a part of Ukraine that is not directly affected by the war.

Method: This is a cross-sectional study conducted in Ternopil. All primary schools in Ternopil were contacted, and children and adolescents aged 7 to 17 years were included in the study if the legal guardian did not object to participation. The Child Revised Impact of Events Scale (CRIES-8) was completed by the youth, whereas demographic characteristics were collected from the youth, parents/legal guardians, and teachers.

Results: A total of 4,753 children and adolescents from 26 schools, aged 7 to 17 years (mean age 11.03 ±1.70 years), 52.3% female, participated in this study. The prevalence of clinically significant PTSD symptoms, defined as CRIES-8 scores of 17 or above, was 40.3% in the overall sample. Female participants had significantly higher intrusion, avoidance, and total CRIES-8 scores (p < 0.001) compared to male participants. The highest CRIES-8 scores were observed in the youngest age group (7-10 years).

Conclusion: The study shows a high prevalence of significant PTSD symptoms even among Ukrainian children and adolescents living in areas far from active conflict zones, suggesting an urgent need for solutions to protect and improve the mental health of youth in war-affected countries.

Plain language summary: The authors studied the mental health of over 4,700 school children in Ternopil, a region of Ukraine far from active fighting. Using a short questionnaire on war-related stress, the authors found that around 40% of them had symptoms that indicated they were at risk of posttraumatic stress disorder (PTSD), especially younger children and girls. The results suggest that war may impact the mental health of children even in relatively safer areas and that widespread accessible psychological support may be urgently needed.

Clinical guidance: • Clinicians should consider screening children from war-affected countries for trauma-related symptoms, even if they live far from active conflictzones, as significant stress may be present without direct exposure.• Younger children may require special attention, as early school-age youth appear to be particularly vulnerable to war-related stress and may exhibitPTSD symptoms differently than adolescents.• Girls may require more targeted monitoring and early intervention, given the consistently higher levels of trauma-related symptoms in this group.

Study registration information: Evaluation of the Teaching Recovery Techniques plus Parenting, a Cluster Randomised Controlled Trial in Ukrainian Schools in Ternopil (TRUST); https://doi.org/10.1186/ISRCTN15476405

Diversity & Inclusion Statement: We worked to ensure sex and gender balance in the recruitment of human participants. We worked to ensure race, ethnic, and/or other types of diversity in the recruitment of human participants. We worked to ensure that the study questionnaires were prepared in an inclusive way. Diverse cell lines and/or genomic datasets were not available. While citing references scientifically relevant for this work, we also actively worked to promote sex and gender balance in our reference list. While citing references scientifically relevant for this work, we also actively worked to promote inclusion of historically underrepresented racial and/or ethnic groups in science in our reference list. The author list of this paper includes contributors from the location and/or community where the research was conducted who participated in the data collection, design, analysis, and/or interpretation of the work.

Timely assessment of regional economic activity in Ukraine is severely constrained by institutional and data-related limitations. Official regional gross regiona lproduct (GRP) statistics are available only at low frequency, are published with substantial delays, and, in the post-2022 period, are further affected by disruptions to statistical production caused by martial law. At the same time, a growing set of potentially informative regional indicators derived from administrative records and official short-term statistics is available at higher frequencies but only over short and heterogeneous time spans. These features make the direct application of standardr egional nowcasting models infeasible.

This paper develops a mixed-frequency factor-augmented vector autoregressive framework tailored to the Ukrainian data environment and designed to incorporate short and incomplete regional indicators into the nowcasting of regional GDP. The model explicitly exploits the hierarchical structure of Ukrainian regional statistics by combining information from quarterly and annual measures of economic activity and by linking regional dynamics to national output developments. Short regional indicators are summarised through latent regional factors extracted using missing-data factor estimation techniques that are robust to ragged edges at both the beginning and the end of the sample.

The proposed framework is implemented using Ukrainian macro-regional aggregates constructed from official data published by the State Statistics Service of Ukraine. Particular attention is paid to the treatment of labour market indicators, housing price dynamics, and other short-term variables that exhibit discontinuities or limited availability. A pseudo-real-time nowcasting exercise shows that conditioning regional GDP nowcasts on factor information derived from short regional data improves predictive performance when contemporaneous national GDP estimates are not yet available. Once national aggregates are released, the marginal informational contribution of regional short-term indicators diminishes.

Overall, the results demonstrate that mixed-frequency factor-augmented VAR models provide a coherent and empirically viable framework for regional GDP now-casting in Ukraine. The approach is particularly well suited to data environments characterised by short samples, publication delays, and institutional disruptions, and thus offers a valuable tool for real-time regional economic monitoring in periods of heightened uncertainty.

The test for the location of the tangency portfolio on the set of feasible portfolios is proposed when both the population and the sample covariance matrices of asset returns are singular. The particular case of investigation is when the number of observations, n, is smaller than the number of assets, k, in the portfolio, and the asset returns are i.i.d. normally distributed with singular covariance matrix Σ such that rank(Σ) = r < n < k + 1. The exact distribution of the test statistic is derived under both the null and alternative hypotheses. Furthermore, the high-dimensional asymptotic distribution of that test statistic is established when both the rank of the population covariance matrix and the sample size increase to infinity so that r/n → c ∈ (0, 1). Theoretical findings are completed by comparing the high-dimensional asymptotic test with an exact finite sample test in the numerical study. A good performance of the obtained results is documented. To get a better understanding of the developed theory, an empirical study with data on the returns on the stocks included in the S&P 500 index is provided.

Modern companies face immense pressure to accelerate and refine decisions related to product assortment due to rapid changes and growing competition in the retail landscape. The volume, velocity, and volatility of business data make intuitive or situational approaches insufficient. Advances in optimization theory and forecasting models enable the design of robust, flexible decision-support systems that bridge the gap between business intuition and data-driven strategy.

In retail, risk manifests primarily through operational inefficiencies, such as capital immobilized in unsold inventory and delayed responsiveness to demand changes. This demands a rethinking of risk modeling tailored specifically to the retail domain.

At the same time, simplistic forecasting tools often prioritize short-term fluctuations at the expense of strategic seasonal trends, thereby undermining long-term planning. As a result, there is a critical need for integrated models that combine predictive accuracy with optimization under uncertainty. Such models must not only capture patterns in consumer demand but also align with operational constraints to ensure that solutions are implementable in practice.

This work proposes a novel, multi-layered framework for assortment optimization that incorporates two key components: SARIMAX-based demand forecasting and the Discrete Functional Particle Method (DFPM) for iterative optimization. Additionally, we introduce a new operational risk measure Inventory Efficiency Ratio (IER) designed to quantify inefficiencies in the retail pipeline.

By embedding these techniques into a unified system, we offer a practical solution for improving capital productivity, reducing inventory holding costs, and enhancing responsiveness in assortment planning. The methodology is validated through real-world data and demonstrates substantial performance improvements over standard planning strategies.

In this paper, we study solvability of the Cauchy problem for a parabolic pseudodifferential equation with the deviation of the argument. Parabolic pseudodiffer-ential operator with non-smooth symbols introduced by Eidel’man and Drin’ for the first time. For such equations, the initial condition is set on a certain interval. Technical and physical reasons for delays can be transport delays, delays in decision-making, delays in information transmission, etc. The most natural are delays when modeling objects in medicine, population dynamics, ecology, etc. Other physical and technical interpretations are also possible, for example, the molecular distribution of thermal energy in various media (liquids, solid bodies, etc.) is modeled by heat conduction equations. Features of the dynamics of vehicles in different environments (water, land, air) can also be taken into account by introducing a delay. The formula for the solution of the Cauchy problem is constructed for the nonlinear equation of heat conduction with a deviation of the argument, its properties are investigated.

Timely assessment of regional economic activity is essential for evidence-based policy and crisis response, particularly in economies where official statistics are released with significant delays. In Ukraine, regional Gross Regional Product (GRP) figures appear only once per year and with a lag of up to 16 months, severely limiting real-time monitoring. This study develops a Mixed-Frequency Factor-Augmented Vector Autoregression (MF-FAVAR) model to nowcast quarterly GRP growth for the Kyiv region by combining annual, quarterly, and monthly indicators.

The proposed framework integrates traditional macroeconomic statistics with high-frequency digital signals derived from Google Trends, capturing shifts in consumer sentiment and behavioural patterns. These digital indicators serve as real-time proxies for household sentiment, consumption intentions, and labour market expectations, offering a complementary perspective to lagged official statistics.

Empirical results demonstrate that EMPCA-based factor extraction delivers the most stable and accurate nowcasts across short, medium, and long horizons. In particular, EMPCA achieves the lowest Root Mean Squared Forecast Error (RMSFE) and Continuous Ranked Probability Score (CRPS), confirming its robustness under sparse and noisy conditions. The findings suggest that mixed-frequency factor-augmented frameworks offer a practical and efficient solution for regional nowcasting under data scarcity, making them especially relevant for transition and crisis economies such as Ukraine.

Своєчасна оцiнка регiональної економiчної активностi є ключовою для прийняття обґрунто-ваних рiшень та реагування на кризовi ситуацiї, особливо в економiках, де офiцiйна статистка публiкується iз суттєвими затримками. В Українi данi про валовий регiональний продукт (ВРП) оприлюднюються лише один раз на рiк iз запiзненням до 16 мiсяцiв, що значно ускладнює монiторинг економiчної ситуацiї в реальному часi. У цьому дослiдженнi запропоновано модель Mixed-Frequency Factor-Augmented Vector Autoregression (MF-FAVAR) для теперiшнього прогнозування (nowcasting) квартального зростання ВРП для київського регiону шляхом поєднання рiчних, квартальних i мiсячних показникiв.

Запропонована структура iнтегрує традицiйнi макроекономiчнi статистичнi показники з високочастотними цифровими сигналами, отриманими з Google Trends, що дає змогу вiдстежувати змiни у споживчих настроях та поведiнкових патернах. Цi цифровi iндикатори виступають проксi змiн у споживчих настроях, намiрах щодо витрат i очiкуваннях на ринку працi, надаючи додаткову iнформацiю порiвняно з офiцiйною статистикою, що публiкується iз затримкою. Зменшення розмiрностi даних здiйснюється за допомогою сучасних методiв факторної екстракцiї, розроблених для неповних i неузгоджених наборiв даних, зокрема Expectation–Maximisation Principal Component Analysis (EMPCA), Bayesian PCA (BPCA) та Singular Value Decomposition Imputation (SVDI). Рiчний ряд ВРП було перетворено на квартальний за допомогою методу Дентона–Шолетта, що забезпечує узгодженiсть з офiцiйними пiдсумками.

Емпiричнi результати показують, що факторна екстракцiя на основi EMPCA забезпечує найстабiльнiшi та найточнiшi коротко-, середньо- та довгостроковi прогнози. Зокрема, EMPCA досягає найменших значень середньоквадратичної похибки прогнозу (RMSFE) та безперервного рангового ймовiрнiсного показника (CRPS), що пiдтверджує її стiйкiсть у умовах обмежених i зашумлених даних. Отриманi результати свiдчать, що моделi зi змiшаною частотою тафакторною структурою є ефективним iнструментом для регiонального nowcasting за умов нестачi даних, що робить їх особливо релевантними для перехiдних i кризових економiк, таких, як Україна.