FURTHER INEQUALITIES FOR THE EUCLIDEAN OPERATOR RADIUS

By use of some non-negative Hermitian forms defined for n-tuple of bounded linear operators on the Hilbert space (H, h·, ·i) we establish new numerical radius and operator norm inequalities for sum of products of operators

On the Reformulated Second Zagreb Index of Graph Operations

Topological indices (TIs) are expressed by constant real numbers that reveal the structure of the graphs in QSAR/QSPR investigation. The reformulated second Zagreb index (RSZI) is such a novel TI having good correlations with various physical attributes, chemical reactivities, or biological activities/properties. The RSZI is defined as the sum of products of edge degrees of the adjacent edges, where the edge degree of an edge is taken to be the sum of vertex degrees of two end vertices of that edge with minus 2. In this study, the behaviour of RSZI under graph operations containing Cartesian product, join, composition, and corona product of two graphs has been established. We have also applied these results to compute RSZI for some important classes of molecular graphs and nanostructures.

CTNI-54. A SINGLE ARM PHASE II STUDY OF THE DUAL MTORC1/MTORC2 INHIBITOR VISTUSERTIB PROVIDED FOR SPORADIC PATIENTS WITH GRADE II-III MENINGIOMAS THAT RECUR OR PROGRESS AFTER SURGERY AND RADIATION

Grade II/III meningiomas have increased rates of recurrence with no approved medical therapies. The historical progression-free survival at 6 months (PFS-6) is 25% with rates >35% declared of interest for drug development. NF2 gene inactivation occurs in about half of meningiomas. Based on our studies showing mTORC1 and mTORC2/SGK1 pathway activation in NF2-deficient meningiomas and the paradoxical activation of the mTORC2/AKT pathway, we hypothesized that mTORC1/mTORC2 inhibitors would be active in meningiomas. We studied the effect of vistusertib in patients with progressive/recurrent grade II/III meningiomas (NCT03071874). Vistusertib was administered orally at 125mg twice daily on two consecutive days each week. MRIs were obtained every 56 days. Tumor size was defined as the largest cross-sectional area. Progression was defined as ≥ 25% increase in the sum of products of all measurable lesions over smallest sum observed. The primary endpoint was PFS-6. Secondary endpoints included toxicity, radiographic response, and correlative studies including immunohistochemistry for mTORC1/2 pathway activation and genetic biomarkers. Twenty-eight patients (13 female, median age 58 years, median KPS 80%) were enrolled. Median tumor size was 4.4cm; 71% were grade II and 50% harbored pathogenic NF2 variants. Four patients discontinued treatment voluntarily and 1 each withdrew for intercurrent illness and non-compliance. PFS-6 is 47% (CI, 26%-65%) and OS-12 is 72% (95%CI, 48%-86%). PFS but not OS was shorter for patients with grade 3 meningiomas; there was no difference in PFS/OS between genetic groups. Adverse events at least possibly related to vistusertib with frequency >10% include nausea, fatigue, hypophosphatemia, diarrhea, anorexia, dry mouth, hypertriglyceridemia, hypertension, vomiting, increased ALT, constipation, and weight loss. Vistusertib treatment was associated with a PFS-6 rate exceeding the target of 35% for recurrent high-grade meningioma. Adverse events were tolerable in this patient population. These data support the continued development of mTORC1/2 inhibitors in this setting.

On the Ternary Exponential Diophantine Equation Equating a Perfect Power and Sum of Products of Consecutive Integers

Consider the Diophantine equation yn=x+x(x+1)+⋯+x(x+1)⋯(x+k), where x, y, n, and k are integers. In 2016, a research article, entitled – ’power values of sums of products of consecutive integers’, primarily proved the inequality n= 19,736 to obtain all solutions (x,y,n) of the equation for the fixed positive integers k≤10. In this paper, we improve the bound as n≤ 10,000 for the same case k≤10, and for any fixed general positive integer k, we give an upper bound depending only on k for n.

Anomalous dimensions of effective theories from partial waves

On-shell amplitude methods have proven to be extremely efficient for calculating anomalous dimensions. We further elaborate on these methods to show that, by the use of an angular momentum decomposition, the one-loop anomalous dimensions can be reduced to essentially a sum of products of partial waves. We apply this to the SM EFT, and show how certain classes of anomalous dimensions have their origin in the same partial-wave coefficients. We also use our result to obtain a generic formula for the one-loop anomalous dimensions of nonlinear sigma models at any order in the energy expansion, and apply our method to gravity, where it proves to be very advantageous even in the presence of IR divergencies.

Use of heurestic methods in marketing modeling

The features of the mechanism of heuristic methods application in marketing modeling are investigated in this paper. The essence of methods of economic analysis in advertising is revealed. Approaches to media planning using heuristic methods are analyzed in detail. Step-by-step evaluation matrix and expert preferences matrix for ranking the advertising evaluation criteria are proposed. The use of integrated models for advertising campaigns modeling is substantiated. Mathematical expressions for their application in the integrated models construction and the use of heuristic methods in media planning are presented. The matrix of expert advantages in terms of ranking the advertising evaluating criteria is proposed. The application of «Delphi» method for processing the results of the analysis of iterative opinions convergence is proposed. The results of modeling the options for evaluating the advertising media choice are presented. It is proved that in the market economy the importance of modeling marketing processes is growing and is that firstly it is necessary to test it on laboratory models, and then in the current market conditions to model the processes required for quality activities in the market. During the crisis most of the companies need strategic forecasting of their activities and for this purpose it is advisable to use methods of economic analysis. It is investigated that heuristic methods (methods of activating creativity, non-standard thinking) include the group of methods for generating solutions of problems. It is determined that the methods of heuristic search are diverse and include both methods of activating creativity and their modification and combination. It is proposed to conduct modeling using heuristic methods in several stages. For this purpose, step-by-step model including thirty steps is used. In this paper expert methods for marketing analysis are used, as they are a set of logical and mathematical techniques, as a result of which experts receive the information necessary for making informed rational management decisions. Both methods of non-directional search including methods of brainstorming, expert assessments, associations and analogies, control questions, collective notebook and directional search, including the algorithm for solving inventive problems are used in this investigation. Heuristic methods are used in marketing analysis because they involve the study and formalization of human thinking in solving various problems. It is proposed in this paper to improve the algorithm for choosing the optimal type of advertising, which is carried out in four stages and has clear heuristic basis. It is offered in this algorithm to include the definition of complex priority in the third stage making it possible to estimate qualitatively the selected options of advertising. The complex indicator for each type of advertising is defined as the sum of products of the relative priorities of advertising on the relative priorities of the criteria for their evaluation. The type of advertising receiving the highest priority can be considered as the best one for the company.

Mappings preserving sum of products a∘b+ba∗ on factor von Neumann algebras

Let A and B be two factor von Neumann algebras. In this paper, we proved that a bijective mapping Φ:A→B satisfies Φ(a∘b+ba∗)=Φ(a)∘Φ(b)+Φ(b)Φ(a)∗ (where ∘ is the special Jordan product on A and B, respectively), for all elements a,b∈A, if and only if Φ is a ∗-ring isomorphism. In particular, if the von Neumann algebras A and B are type I factors, then Φ is a unitary isomorphism or a conjugate unitary isomorphism.

Matrix Approach To The Direct Computation Method For The Solution of Fredholm Integro-Differential Equations of The Second Kind With Degenerate Kernels

In this paper, a matrix approach to the direct computation method for solving Fredholm integro-differential equations (FIDEs) of the second kind with degenerate kernels is presented. Our approach consists of reducing the problem to a set of linear algebraic equations by approximating the kernel with a finite sum of products and determining the unknown constants by the matrix approach. The proposed method is simple, efficient and accurate; it approximates the solutions exactly with the closed form solutions. Some problems are considered using maple programme to illustrate the simplicity, efficiency and accuracy of the proposed method.