Recent trends in portfolio management emphasize the importance of reducing carbon footprints and aligning investments with sustainable practices. This paper introduces Sensitivity Value-at-Risk (SensitivityVaR), an advanced distortion risk measure that combines Value-at-Risk (VaR) and Expected Shortfall (ES) with the Cornish–Fisher expansion. SensitivityVaR provides a more robust framework for managing risk, particularly under extreme market conditions. By incorporating first- and second-order distorted stochastic dominance criteria, we enhance portfolio decarbonization strategies, aligning financial objectives with environmental targets such as the Paris Agreement’s goal of a 7% annual reduction in carbon intensity from 2019 to 2050. Our empirical analysis evaluates the impact of integrating carbon intensity data—including Scope 1, Scope 2, and Scope 3 emissions—on portfolio optimization, focusing on key sectors like technology, energy, and consumer goods. The results demonstrate the effectiveness of SensitivityVaR in managing both risk and environmental impact. The methodology led to significant reductions in carbon intensity across different portfolio configurations, while preserving competitive risk-adjusted returns. By optimizing tail risks and limiting exposure to carbon-intensive assets, this approach produced more balanced and efficient portfolios that aligned with both financial and sustainability goals. These findings offer valuable insights for institutional investors and asset managers aiming to integrate climate considerations into their investment strategies without compromising financial performance.

In the continuous setting, Morrey spaces have been studied extensively, especially since the late 1960s. Meanwhile, Morrey sequence spaces, which are also known as discrete Morrey spaces, have only been developed by Gunawan et al. since 2018. In this article, we extend some known results on their inclusion properties and their (lack of) uniform nonsquareness to mixed Morrey double-sequence spaces, i.e. Morrey double-sequence spaces equipped with a mixed norm. As in the calculation of three geometric constants of Morrey spaces by Gunawan et al. in 2019, we also compute three geometric constants, namely Von Neumann-Jordan constant, James constant, and Dunkl-Williams constant for mixed Morrey double-sequence spaces. These constants measure uniformly nonsquareness of any Banach space. Through the values of the three constants, we reveal that mixed Morrey double-sequence spaces are not uniformly nonsquare. A relation between mixed Morrey double-sequence spaces and mixed Morrey spaces is also discussed.

In this paper, several extensions of the isoperimetric problem in solid figures are explored, focusing on oblique and right prisms with rectangular, right-angled triangular, and regular hexagonal bases. The objective of this research is to find the prism with the largest volume while keeping the surface area constant. Through manipulations of algebra and simple trigonometry, evidence is obtained that a right prism provides a larger volume than an oblique prism if their surface areas are equal. By utilizing partial derivatives of a two-variable function and the Lagrange multiplier method, conditions for the side lengths are derived to obtain the prism with the maximum volume. The results show that a cube is the solution to the isoperimetric problem, meaning it has the largest volume among prisms with rectangular bases, while for the isoperimetric solution on prisms with right-angled triangular bases, the base of the prism must be an isosceles right-angled triangle. A regular hexagonal prism has a larger volume than prisms with rectangular and right-angled triangular bases if their surface areas are the same.

Dependent Tail Value-at-Risk, abbreviated as DTVaR, is a copula-based extension of Tail Value-at-Risk (TVaR). This risk measure is an expectation of a target loss once the loss and its associated loss are above their respective quantiles but bounded above by their respective larger quantiles. In this paper, we propose nonparametric estimators for DTVaR and establish their property of consistency. Moreover, we also propose the variability measure around this expected value truncated by the quantiles, called the Dependent Conditional Tail Variance (DCTV). We use this measure for constructing confidence intervals of the DTVaR. Both parametric and nonparametric approaches for DTVaR estimations are explored. Furthermore, we assess the performance of DTVaR estimations using a proposed backtest based on the DCTV. As for the numerical study, we take an application in the insurance claim amount.

For integer k ≥ 2, let X = {0, 1, 2, …, k}. In this paper, we determine the order of a star graph K1, n of n + 1 vertices, such that K1, n admits a topological integer additive set-labeling (TIASL) with respect to a set X. We also give a condition for a star graph K1, n such that K1, n is not a TIASL-graph on set X.

The concept of n-normed spaces is a generalization of the concept of normed spaces. Some characteristics of n-normed spaces have been discussed by many researchers. These spaces are usually observed using a set of n linear independent vectors. In this paper, we will construct quotient spaces of an n-normed space with respect to n linear independent vectors. We define a norm in each quotient space by using the n-norm that we have. Norms of these quotient spaces will be a new viewpoint in observing characteristics of n-normed spaces.

In isotropic semivariograms, ordinary least squares can estimate nugget effect and sill by partitioning its range. By conducting simulation, a semivariogram model with previously given parameters will be estimated through bootstrap method. Least square-bootstrap (LS-Bootstrap) will be applied to estimate the parameters of the model after resampling the errors of the model. The selection of the resulting semivariogram model from bootstrap method will be affected by the number of distance lags, the precision level of the range partitions, the number of bootstrap iterations, and the given reference model. The exponential and Gaussian models are sufficiently good in the estimation for the models with the same references. Meanwhile, the estimation yielded from spherical model is quite far from the reference 5124 K. N. Sari et al. exponential and Gaussian models, with the mean square error value reaching 713. The estimation with bootstrap method which is the same as the reference model will be faster to converge with the maximum iteration of 50. Besides, bootstrap method enables to obtain the point estimates and interval estimates of the nugget effect, sill, and range parameters.

In this paper we prove fixed point theorems for contraction mappings and phi-contraction mappings on a bounded and closed set with respect to n linearly independent vectors in an n-normed space. Our results rectify those obtained recently by Kir and Kiziltunc [11].

Anisotropic semivariogram modeling can be aplied in petroleum industry where the angle between a pair of wells has important function in defining the spatial correlation between wells. In geometry anisotropic, function of range is formulated in trigonometric functions of the angle between pairs of wells that have periodicity property. The fluctuations of range will affect on shifting geometry anisotropic models with different properties for each quadrant of angle. In three semivariogram models (exponential, spherical and gaussian), the increasing of angle give difference influence for range function and the shifting of semivariogram value.

This paper provides an application of generalized space-time autoregressive (GSTAR) model on GDP data in West European countries. Preliminary model is identified by space-time ACF and space-time PACF of the sample, and model parameters are estimated using the least square method. The forecast performance is evaluated using the mean of squared forecast errors (MSFEs) based on the last ten actual data. It is found that the preliminary model is GSTAR(2;1,1). As a comparison, the estimation and the forecast performance are also applied to the GSTAR(1;1) model which has fewer parameter. The results showed that the ASFE of GSTAR(2;1,1) is smaller than that of the order (1;1). However, the t-test value shows that the performance is significantly indifferent. Thus, due to the parsimony principle, the GSTAR(1;1) model might be considered as a forecasting model.

In this paper we discuss the concept of n-normed spaces. In particular, we show the equality of four different formulas of n-norms in a Hilbert space. In addition, we study the notion of bounded n-linear functionals on an n-normed space and present some results on it.

The notion of angles is known in a vector space equipped with an inner product, but not well established in a vector space equipped only with a norm. In this note, we shall develop some notions of angles between two vectors in a normed space and discuss their properties.

We present an explicit formula for angles between two subspaces ofinner product spaces. Our formula serves as a correction for, as well as an extensionof, the formula proposed by Risteski and Trenˇcevski [13]. As a consequence of ourformula, a generalized Cauchy-Schwarz inequality is obtained.MSC 2000: 15A03, 51N20, 15A45, 15A21, 46B20Keywords: Angles between subspaces, canonical angles, generalized Cauchy-Schwarz inequality1. IntroductionThe notion of angles between two subspaces of the Euclidean space R d has been studiedby many researchers since the 1950’s or even earlier (see [3]). In statistics, canonical (orprincipal) angles are studied as measures of dependency of one set of random variables onanother (see [1]). Some recent works on angles between subspaces and related topics canbe found in, for example, [4, 8, 12, 13, 14]. Particularly, in [13], Risteski and Trenˇcevskiintroduced a more geometrical definition of angles between two subspaces of R d and explainedits connection with canonical angles. Their definition of the angle, however, is based on ageneralized Cauchy-Schwarz inequality which we found incorrect. The purpose of this noteis to fix their definition and at the same time extend the ambient space to any real innerproduct space.Let (X,h·,·i) be a real inner product space, which will be our ambient space throughoutthis note. Given two nonzero, finite-dimensional, subspaces U and V of X with dim(U) ≤