Instability, index theorem, and exponential trichotomy for Linear Hamiltonian PDEs

Consider a general linear Hamiltonian system ∂ t u = J L u \partial _{t}u=JLu in a Hilbert space X X . We assume that L : X → X ∗ \ L:X\rightarrow X^{\ast } induces a bounded and symmetric bi-linear form ⟨ L ⋅ , ⋅ ⟩ \left \langle L\cdot ,\cdot \right \rangle on X X , which has only finitely many negative dimensions n − ( L ) n^{-}(L) . There is no restriction on the anti-self-dual operator J : X ∗ ⊃ D ( J ) → X J:X^{\ast }\supset D(J)\rightarrow X . We first obtain a structural decomposition of X X into the direct sum of several closed subspaces so that L L is blockwise diagonalized and J L JL is of upper triangular form, where the blocks are easier to handle. Based on this structure, we first prove the linear exponential trichotomy of e t J L e^{tJL} . In particular, e t J L e^{tJL} has at most algebraic growth in the finite co-dimensional center subspace. Next we prove an instability index theorem to relate n − ( L ) n^{-}\left ( L\right ) and the dimensions of generalized eigenspaces of eigenvalues of J L \ JL , some of which may be embedded in the continuous spectrum. This generalizes and refines previous results, where mostly J J was assumed to have a bounded inverse. More explicit information for the indexes with pure imaginary eigenvalues are obtained as well. Moreover, when Hamiltonian perturbations are considered, we give a sharp condition for the structural instability regarding the generation of unstable spectrum from the imaginary axis. Finally, we discuss Hamiltonian PDEs including dispersive long wave models (BBM, KDV and good Boussinesq equations), 2D Euler equation for ideal fluids, and 2D nonlinear Schrödinger equations with nonzero conditions at infinity, where our general theory applies to yield stability or instability of some coherent states.

Growth, Instability and Decomposition Analysis of Nutri Cereals in Tamil Nadu

Aim: The study was undertaken to know the growth rate, instability, and contribution of area and yield on the production of Nutri cereals. Place and Duration: The study was based on the secondary data for the period of 20 years (2001- 2020) for Tamil Nadu. Data has been collected from Agricultural statistics at a glance (2020), Season and Crop report (2020). Methodology: The growth rate of Nutri cereals can be estimated through compound annual growth rate, instability is measured using Coppock’s instability index and contribution of area and yield on production is studied using decomposition analysis. Results: Results revealed that Nutri cereals growth rate is found to be negative in first decade and positive in second decade Production of Nutri cereals in both the decades found positive in decline trend. The trend of productivity seems positive for both the decades at declining rate. Instability index is higher in terms of production than area and productivity. Area effect was most responsible for the production of Nutri cereals than yield and interaction effect. Conclusion: The area effect is more responsible for production of Nutri cereals. Area effect on production of Nutri cereals is high, hence government should intervene in this regard to improve the area of Nutri cereals cultivation.

Modified instability index of the troposphere associated with thunderstorms / nor'westers over Bangladesh during the pre-monsoon season

The present study is an attempt to study different Modified Stability Indices in relation to the occurrence of severe thunderstorms/nor’westers in order to find out the critical values of different modified indices favorable for the formation of thunderstorms in Bangladesh. Computations have also been made for the stations in and around Bangladesh for studying the spatial distribution of the modified stability indices. The Modified Instability Indices such as Modified Cross Total Index (MCT), Modified Vertical Total Index (MVT), Modified Total Totals Index (MTT), Modified SWEAT Index (MSWI), Modified K-Index (MKI) and Modified Energy Index (MEI) show greater instability of the troposphere in the morning as compared to CT, VT, TT, SWI, KI and EI. The critical values of different modified instability indices at 0000 UTC over Dhaka are: MCT >= 20° C, MVT >= 26° C, MTT >= 46° C, MSWI > 300, MKI >= 40° C and MEI < -6 joules/gm respectively for the nor’westers to occur in Bangladesh. The spatial distributions of modified stability indices have revealed that maximum instability lies over the area of surface low pressure especially over Bihar, West Bengal and adjoining Bangladesh at 0000 UTC on the dates of occurrence of nor’westers. Nor’westers occur at theeastern end of the maximum instability. For severe nor’westers of tornadic intensity, the critical values of different modified instability indices at 0000 UTC over Dhaka are: MCT >= 20° C, MVT >= 28° C, MTT >= 50° C, MSWI >= 500, MKI >= 42° C and MEI < -8 joules/gm respectively for the nor’westers to occur in Bangladesh.

Medium levels of transcription and replication related chromosomal instability are associated with poor clinical outcome

Genomic instability (GI) influences treatment efficacy and resistance, and an accurate measure of it is lacking. Current measures of GI are based on counts of specific structural variation (SV) and mutational signatures. Here, we present a holistic approach to measuring GI based on the quantification of the steady-state equilibrium between DNA damage and repair as assessed by the residual breakpoints (BP) remaining after repair, irrespective of SV type. We use the notion of Hscore, a BP “hotspotness” magnitude scale, to measure the propensity of genomic structural or functional DNA elements to break more than expected by chance. We then derived new measures of transcription- and replication-associated GI that we call iTRAC (transcription-associated chromosomal instability index) and iRACIN (replication-associated chromosomal instability index). We show that iTRAC and iRACIN are predictive of metastatic relapse in Leiomyosarcoma (LMS) and that they may be combined to form a new classifier called MAGIC (mixed transcription- and replication-associated genomic instability classifier). MAGIC outperforms the gold standards FNCLCC and CINSARC in stratifying metastatic risk in LMS. Furthermore, iTRAC stratifies chemotherapeutic response in LMS. We finally show that this approach is applicable to other cancers.

Exploring Life Instability’s Relationship to the Mental Health of Older Adults With HIV

The study is one of the first to examine both the prevalence of life instability among older adults with HIV (OAWH) in a community clinic and its relationship to their mental health. OAWH (N=623) from a community medical clinic completed an interviewer-administered assessment (English/Spanish) which included an additive Life Instability Index (LII) composed of indicators at the individual (e.g. education, housing instability, employment status) and community (e.g. poverty, transportation) levels. Participants were a mean age of 60 years (SD = 5.90) with the majority identifying as Black-non-Hispanic (65.9%), cisgender male (60.8%), and heterosexual (80.6%). Participants reported an average of 6.08 destabilizing factors (SD = 1.44). In multiple linear regression analyses LII was significantly related to increased substance use among participants (b= 0.08, p &lt; 0.01), but not with anxiety or depression. An LII is an innovative approach to assess the relationship between OAWH’s mental health and social determinants of health.

Analytical Study of Kharif Food Grain Production in Odisha

The state of Odisha having an agrarian based economy depends largely on agriculture for the livelihood of its population. Food grains are important commodity of crop groups that provide high quality carbohydrates, protein and vitamins. A study on the compound growth rate of area, yield and production of food grains for kharif season in the districts of Odisha and the state as a whole has been attempted in the present study which would be helpful in visualizing the progress of the state with respect to food grain cultivation and proper framing of agricultural policies of the state. The study is based secondary data for the period of 1993-94 to 2017-18 to estimate the compound growth rate and Cuddy-Della Instability Index of area, yield and production of kharif food grains for the districts and the state as a whole. The districts are ranked on the basis of compound growth rate and Cuddy-Della Instability Index in decreasing order and increasing order of their magnitudes respectively. The rank correlation between Compound Growth Rate and Cuddy-Della Instability Index of area, yield and production of food grains during kharif seasons are studied. It is found that despite negative growth rate in area, the positive compound growth rate of yield leads to positive compound growth rate in production of kharif food grains of Odisha. Also it is found that despite stability of area, the instability of yield leads to instability in production of kharif food grains in the state of Odisha.

Fatores prognósticos da cirurgia de Latarjet

Resumo Objetivo Investigar os fatores que influenciam os resultados funcionais dos pacientes submetidos ao procedimento de Latarjet. Métodos Avaliação de 26 pacientes submetidos ao tratamento cirúrgico, seguindo a técnica de Latarjet, devido a luxação anterior recidivante traumática da articulação glenoumeral, com perda óssea glenoidal maior do que 20% e/ou lesão off-track. O tempo mínimo de seguimento foi de doze meses. Foram avaliadas as seguintes escalas: Escala Visual Analógica (EVA), The Western Ontario Shoulder Instability Index (WOSI), e Subjective Shoulder Value (SSV), além dos dados objetivos dos participantes. Resultados A maioria dos pacientes (84,62%) não apresentou recidiva de luxação, e 92,31% ficaram satisfeitos. Em relação à análise funcional, a pontuação do componente físico (PCF) e a pontuação do compenente mental (PCM) encontradas estavam dentro da média de qualidade de vida da população. Os sintomas físicos, pela escala WOSI, apresentaram a melhor porcentagem (8,5%), ao passo que o pior resultado foi observado com relação ao estilo de vida (20%). Na EVA, a dor foi classificada como moderada (3/10) por 15,38% dos pacientes (4/26). Em relação aos esportes, os pacientes que praticavam alguma atividade esportiva apresentaram melhora nos parâmetros da escala SSV, que tiveram relação inversa com o número de recidivas. Observou-se ainda que, quanto menor foi o tempo entre a primeira luxação e a realização da cirurgia, maior foi a satisfação do paciente. Conclusão A indicação precoce do tratamento cirúrgico da instabilidade anterior glenoumeral pode proporcionar melhores resultados funcionais subjetivos ao paciente.

Trend and Instability Index Analysis in Paddy Crop Area, Productivity and Production across District in Andhra Pradesh, India

Paddy is the important, food grain, cereal crop, and staple food crop of India. India ranks second in production of paddy after China. This establishes the pressing need for the study of vulnerability and trend of paddy cultivation and productivity in India. Andhra Pradesh being one of the largest producers of paddy, the compound growth rate and instability in area, productivity and production of paddy in Andhra Pradesh was assessed after the remainder of the state of undivided Andhra Pradesh after bifurcation in 2014 with the jurisdiction of four Rayalaseema and nine Coastal Andhra districts. In this study, we have analyzed the trend & carried out instability index analysis in paddy crop area, productivity and production across district of Andhra Pradesh to explicate the trend in extent paddy cultivation and paddy productivity, district wise Cuddy-Della Valle-Instability Index (%) for the paddy cultivating area from 1991-92 to 2018-19, using the data collected from Directorate of Economics and Statistics, Andhra Pradesh. We assessed district wise Cuddy-Della Valle-Instability Index (%) in paddy productivity from 1991-92 to 2018-19 and found that in Andhra Pradesh, the area of paddy cultivation, paddy productivity output has changed in the study area over the time. Between 1991-92 and 2018-19, the overall paddy growth rate of the area showed a decline of 0.5%. And growth rate of paddy productivity was 1.31% while the increase of production was 0.74%, which can be attributed to the negative impacts arising out of changing weather patterns.

Performance of Millets in Bundelkhand Region of UP State

Time series data on area, production and productivity of major millets (2000-01 to 2019- 20) was analyzed for Bundelkhand region of Uttar Pradesh state. Sorghum was the most important millet crop that accounted for highest area (36.6%) and (34%) of total millet area and production in UP state, respectively. Chitrakoot district contributed highest area (31.6%) under millets. Among all the millets, pearl millet recorded highest growth rate in area (0.97%), production (3.57%) and productivity (1.59%) with low instability index for area. The overall area and production of millet crops recorded decline in all the seven districts of Bundelkhand region during overall study period (2000-20). Lalitpur district recorded highest decline in area (-22.02%), followed by Chitrakoot (-10.82%) and Jhansi district (- 10.63%) during overall study period, while during the same period Banda district recorded a growth in area (1.16%) and production (4.77 %) of pearl millet. The overall area, production and productivity of millets registered a decline in Bundelkhand region during 2000-20. This calls for aggressive promotional activities for enhancing millets production in the region.

Counting equilibria of large complex systems by instability index

We consider a nonlinear autonomous system of N≫1 degrees of freedom randomly coupled by both relaxational (“gradient”) and nonrelaxational (“solenoidal”) random interactions. We show that with increased interaction strength, such systems generically undergo an abrupt transition from a trivial phase portrait with a single stable equilibrium into a topologically nontrivial regime of “absolute instability” where equilibria are on average exponentially abundant, but typically, all of them are unstable, unless the dynamics is purely gradient. When interactions increase even further, the stable equilibria eventually become on average exponentially abundant unless the interaction is purely solenoidal. We further calculate the mean proportion of equilibria that have a fixed fraction of unstable directions.