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In 1975, Lieb and Thirring derived a semiclassical lower bound on the kinetic energy for fermions, which agrees with the Thomas-Fermi approximation up to a constant factor. Whenever the optimal constant in their bound coincides with the semiclassical one is a long-standing open question. We prove an improved bound with the semiclassical constant and a gradient error term which is of lower order., 6 pages, comments and references added

We perform a detailed analysis of the solvability of linear strain equations on several non-characteristic regions to obtain L 2 regularity solutions. As an application, the rigidity results on the strain tensor of the middle surface are implied by the L 2 regularity for non-characteristic regions. Finally, we obtain the optimal constant in the first Korn inequality scales like h 4 / 3 for hyperbolic shells, generalizing the assumption that the middle surface of the shell is given by a single principal system in the literature.

Gogatishvili, Amiran/0000-0003-3459-0355; Yildiz, Tugce Unver/0000-0003-0414-8400; Gogatishvili, Amiran/0000-0003-3459-0355; Mustafayev, Rza/0000-0002-2806-9646 WOS: 000416445500016 In this paper, characterizations of the embeddings between weighted Copson function spaces Cop(p1,q1)(u(1),v(1)) and weighted Cesaro function spaces Ces(p2,q2) (u(2) , v(2)) are given. In particular, two-sided estimates of the optimal constant c in the inequality (integral(infinity)(0) (integral(t)(0) f(tau)(p2)v2(tau)d tau)(q2/p2) u2(t)dt)(1/q2)& para;& para

Zero-difference balanced (ZDB) function plays an important role in communication field. In this paper, we propose a generic method to construct ZDB functions on generic algebraic rings. Using this method, we construct many new ZDB functions and retrieve some existing ZDB functions in a much simpler way. Moreover, new applications of the constructed ZDB functions, such as constructing optimal constant weight codes and optimal frequency-hopping sequences, are presented.

Let 0 1p>1 be such that ps0S≡S(N,p,s,β)>0. (3) If β≡N−ps2, as a consequence of the improved Hardy inequality, we obtain that for all q

We analyze how inventory dependence of demand affects the optimal pricing and distribution of inventory across multiple retail locations. Our problem formulation corresponds to the typical markdown management decisions in a retail chain. The solution methodology determines the optimal combinations of three important operational levers—pricing, inventory allocation, and store consolidation. Inventory dependence causes the optimal price trajectory to decrease over time as the current on-hand inventory decreases. The optimal inventory allocation shifts more inventory to the larger stores as the inventory effect becomes stronger. Solution methodologies are also developed for the case of optimal constant prices, which leads to the same optimal inventory allocations. If all stores have the same seasonal variations and the inventory is allocated optimally, then all stores have identical price trajectories. Optimal consolidation of the inventory to a subset of the stores is also determined. Consolidation of inventory to a subset of stores can be beneficial even when there are no fixed costs of stocking a store. Using some typical parameter values for retail items, our calculations show that significant benefits can result from taking the inventory effect into account in pricing, inventory allocation, and store consolidation decisions. The online appendix is available at https://doi.org/10.1287/msom.2016.0609 .

In this paper, we investigate the complete weight enumerators of two classes of irreducible cyclic codes. We present the explicit complete weight enumerator of the irreducible cyclic codes. Furthermore, we obtain a class of optimal constant composition codes from irreducible cyclic codes.

This paper shows the potential of optimal control for enhancing the power capture for a three-body hinge-barge wave energy converter (WEC). It is the first documented application of a coordinated control strategy for a hinge-barge WEC, where power is converted at the joints between both the fore and aft pontoons, and the central barge section. Two separate optimal control formulations, based on different model representations, are evaluated and are shown to considerably outperform an optimal constant damping control strategy.

Let $D \subset \mathbb{C}$ be a domain with $0 \in D$. For $R>0$, let ${{\hat \omega }_D}\left( {R} \right)$ denote the harmonic measure of $ D \cap \left\{ {\left| z \right| = R} \right\}$ at $0$ with respect to the domain $ D \cap \left\{ {\left| z \right| < R} \right\} $ and ${\omega_D}\left( {R} \right)$ denote the harmonic measure of $\partial D \cap \left\{ {\left| z \right| \ge R} \right\}$ at $0$ with respect to $D$. The behavior of the functions ${\omega_D}$ and ${{\hat \omega }_D}$ near $\infty$ determines (in some sense) how large $D$ is. However, it is not known whether the functions ${\omega_D}$ and ${{\hat \omega }_D}$ always have the same behavior when $R$ tends to $\infty$. Obviously, ${\omega_D}\left( {R} \right) \le {{\hat \omega }_D}\left( {R} \right)$ for every $R>0$. Thus, the arising question, first posed by Betsakos, is the following: Does there exist a positive constant $C$ such that for all simply connected domains $D$ with $0 \in D$ and all $R>0$, \[{\omega_D}\left( {R} \right) \ge C{{\hat \omega }_D}\left( {R} \right)? \] In general, we prove that the answer is negative by means of two different counter-examples. However, under additional assumptions involving the geometry of $D$, we prove that the answer is positive. We also find the value of the optimal constant for starlike domains., Comment: 22 pages, 13 figures

The main results in the paper are the weighted multipolar Hardy inequalities \begin{document}$ \begin{equation*} c\int_{\mathbb{R}^N}\sum\limits_{i = 1}^n\frac{\varphi^2}{|x-a_i|^2}\,\mu(x)dx \leq\int_{\mathbb{R}^N}|\nabla \varphi |^2\mu(x)dx+ K\int_{\mathbb{R}^N} \varphi^2\mu(x)dx, \end{equation*} $\end{document} in \begin{document}$ \mathbb{R}^N $\end{document} for any \begin{document}$ \varphi $\end{document} in a suitable weighted Sobolev space, with \begin{document}$ 0 , \begin{document}$ a_1,\dots,a_n\in \mathbb{R}^N $\end{document} , \begin{document}$ K $\end{document} constant. The weight functions \begin{document}$ \mu $\end{document} are of a quite general type. The paper fits in the framework of Kolmogorov operators defined on smooth functions \begin{document}$ \begin{equation*} Lu = \Delta u+\frac{\nabla \mu}{\mu}\cdot\nabla u, \end{equation*} $\end{document} perturbed by multipolar inverse square potentials, and related evolution problems. Necessary and sufficient conditions for the existence of exponentially bounded in time positive solutions to the associated initial value problem are based on weighted Hardy inequalities. For constants \begin{document}$ c $\end{document} beyond the optimal Hardy constant \begin{document}$ c_{o,\mu} $\end{document} we are able to show nonexistence of positive solutions.

Carrying on the discussion initiated in (Dovetta-Tentarelli'18), we investigate the existence of ground states of prescribed mass for the $L^2$-critical NonLinear Schr\"odinger Equation (NLSE) on noncompact metric graphs with localized nonlinearity. Precisely, we show that the existence (or nonexistence) of ground states mainly depends on a parameter called reduced critical mass, and then we discuss how the topological and metric features of the graphs affect such a parameter, establishing some relevant differences with respect to the case of the extended nonlinearity studied by (Adami-Serra-Tilli'17). Our results rely on a thorough analysis of the optimal constant of a suitable variant of the $L^2$-critical Gagliardo-Nirenberg inequality., Comment: 22 pages, 7 figures. Keywords: metric graphs, NLS, ground states, localized nonlinearity, $L^2$-critical case. Some minor revisions have been made with respect to the previous version. Accepted for publication by Calc. Var. Partial Differential Equations

To study the corrosion degradation of cable wires in a bridge&rsquo, s life, this research work created an accelerated corrosion test device, which sought to identify an optimal constant strain level. An accelerated corrosion test was carried out and the corroded specimens were scanned using super depth 3D microscopy technology. Mass loss and minimum cross-sectional diameter was measured to understand the degradation characteristics of cable wires at variable strains and corrosion time. The variation of elastic modulus, yield load, and ultimate load of corroded wires, subjected to a tensile test, were analyzed. The experimental results illustrate that the average mass loss ratio of the corroded cable wires increases nonlinearly as corrosion time increases. The higher the stress level, the more serious the corrosion level. The minimum cross-sectional diameter has good correlation with corrosion time and stress level. The elastic modulus of wires does not change significantly with the increase of corrosion time. Yield load and ultimate load decreases with the increase of strain level, and the rates of decline under different strains are nonlinear.

In this paper we provide an explicit formula for the optimal lower bound of Donaldson's J-functional, in the sense of finding explicitly the optimal constant in the definition of coercivity, which always exists and takes negative values in general. This constant is positive precisely if the J-equation admits a solution, and the explicit formula has a number of applications. First, this leads to new existence criteria for constant scalar curvature K\"ahler (cscK) metrics in terms of Tian's alpha invariant. Moreover, we use the above formula to discuss Calabi dream manifolds and an analogous notion for the J-equation, and show that for surfaces the optimal bound is an explicitly computable rational function which typically tends to minus infinity as the underlying class approaches the boundary of the K\"ahler cone, even when the underlying K\"ahler classes admit cscK metrics. As a final application we show that if the Lejmi-Sz\'ekelyhidi conjecture holds, then the optimal bound coincides with its algebraic counterpart, the set of J-semistable classes equals the closure of the set of uniformly J-stable classes in the K\"ahler cone, and there exists an optimal degeneration for uniform J-stability., Comment: Final version, to appear in Adv. Math

We prove and give numerical results for two lower bounds and eleven upper bounds to the optimal constant k0 = k0(n,α) in the inequality ∥u∥2n/(n-2α) ≤ k0 ∥∇u∥α2 ∥u∥1-α 2, u ∈ H1(ℝn), for n = 1, 0 < α ≤ 1/2, and n ≥ 2, 0 < α < 1. This constant k0 is the reciprocal of the infimum λn,α for u ∈ H1(ℝn) of the functional Λn,α = ∥∇u∥α2 ∥u∥1-α 2/∥u∥2n/(n-2α), u ∈ H1(ℝn), where for n = 1, 0 < α ≤ 1/2, and for n ≥ 2, 0 < α < 1. The lowest point in the point spectrum of the Schrödinger operator τ = -Δ+q on ℝn with the real-valued potential q can be expressed in λn,α for all q _ = max(0,-q) ∈ Lp(ℝn), for n = 1, 1 ≤ p < ∞, and n ≥ 2, n/2 < p < ∞, and the norm ∥q _ ∥p.

In 1989 Almgren and Lieb proved a rearrangement inequality for the Sobolev spaces of fractional order $W^{s,p}$. The case $p = 2$ of their result implies the nonlocal isoperimetric inequality \[ \frac{P_s(E)}{|E|^{\frac{N-2s}N}} \ge \frac{P_s(B_1)}{|B_1|^{\frac{N-2s}N}},\ \ \ \ \ \ \ 0, Comment: This submission represents an entirely revised version, with different title and abstract, of a previous submission, entitled:"Observations on a theorem of Almgren and Lieb"

We focus on the study of ground-states for the system of M coupled semilinear Schrodinger equations with power-type nonlinearities and couplings. General results regarding existence and characterization are derived using a variational approach. We show the usefulness of such a characterization in several particular cases, including those for which uniqueness of ground-states is already known. Finally, we apply the results to find the optimal constant for the vector-valued Gagliardo–Nirenberg inequality and we study global existence, L 2 -concentration phenomena and blowup profile for the evolution system in the L 2 -critical power case.

Partitioned difference families are an interesting class of discrete structures which can be used to derive optimal constant composition codes. There have been intensive researches on the construction of partitioned difference families. In this paper, we consider the combinatorial approach. We introduce a new combinatorial configuration named partitioned relative difference family, which proves to be very powerful in the construction of partitioned difference families. In particular, we present two general recursive constructions, which not only include some existing constructions as special cases, but also generate many new series of partitioned difference families. As an application, we use these partitioned difference families to construct several new classes of optimal constant composition codes.

The isolation of T cells, followed by differentiation into Regulatory T cells (Tregs), and re-transplantation into the body has been proposed as a therapeutic option for inflammatory bowel disease. A key requirement for making this a viable therapeutic option is the generation of a large population of Tregs. However, cytokines in the local microenvironment can impact the yield of Tregs during differentiation. As such, experimental design is an essential part of evaluating the importance of different cytokine concentrations for Treg differentiation. However, currently only single, constant concentrations of the cytokines have been investigated. This work addresses this point by performing experimental design in silico which seeks to maximize the predicted induction of Tregs relative to Th17 cells, by selecting an optimal input function for the concentrations of TGF-β, IL-2, IL-6, and IL-23. While this approach sounds promising, the results show that only marginal improvements in the concentration of Tregs can be achieved for dynamic cytokine profiles as compared to optimal constant concentrations. Since constant concentrations are easier to implement in experiments, it is recommended for this particular system to keep the concentrations constant where IL-6 should be kept low and high concentrations of TGF-β, IL-2, and IL-23 should be used.

We revisit the celebrated family of BDG-inequalities introduced by Burkholder, Gundy (Acta Math. 124 (1970) 249–304) and Davis (Israel J. Math. 8 (1970) 187–190) for continuous martingales. For the inequalities $\mathbb{E}[\tau^{\frac{p}{2}}]\leq C_{p}\mathbb{E}[(B^{*}(\tau))^{p}]$ with $0

We obtain the sharp factor of the two-sides estimates of the optimal constant in generalized Hardy's inequality with two general Borel measures on $\mathbb{R}$, which generalizes and unifies the known continuous and discrete cases.

Let P : C n → C be an m-homogeneous polynomial given by P ( x ) = ∑ 1 ≤ j 1 ≤ … ≤ j m ≤ n c j 1 … j m x j 1 … x j m . Defant and Schluters defined a non-symmetric associated m-form L P : ( C n ) m → C by L P ( x ( 1 ) , … , x ( m ) ) = ∑ 1 ≤ j 1 ≤ … ≤ j m ≤ n c j 1 … j m x j 1 ( 1 ) … x j m ( m ) . They estimated the norm of L P on ( C n , ‖ ⋅ ‖ ) m by the norm of P on ( C n , ‖ ⋅ ‖ ) times a ( c log ⁡ n ) m 2 factor for every 1-unconditional norm ‖ ⋅ ‖ on C n . A symmetrization procedure based on a card-shuffling algorithm which (together with Defant and Schluters' argument) brings the constant term down to ( c m log ⁡ n ) m − 1 is provided. Regarding the lower bound, it is shown that the optimal constant is bigger than ( c log ⁡ n ) m / 2 when n ≫ m . Finally, the case of l p -norms ‖ ⋅ ‖ p with 1 ≤ p 2 is addressed.

In this paper we analyze the tone reservation method to reduce the peak to average power ratio (PAPR) in orthogonal frequency division multiplexing (OFDM) systems. We consider the case of a reduced compensation set, where only the positive carrier frequencies are used, and fully characterize the information sets for which the PAPR problem is solvable. It turns out that the reduction of the compensation set does not affect the solvability of the PAPR problem, however, the optimal constant are worse in general.

Under a correlation constraint the optimal constant/fixed-mix portfolio consists of the market portfolio, the riskless bond and the benchmark.

A new family of effectively nine stages, ninth-order hybrid explicit Numerov-type methods is presented for the solution of some special second order Initial Value Problem. After dealing with a reduced set of order conditions, we derive an optimal constant coefficients method along with a similar kind of method with reduced phase errors. We proceed with numerical tests using quadruple precision arithmetic on some well-known problems from the relevant literature. Finally, in the appendices, we list Mathematica packages implementing the corresponding algorithms.

We consider the minimizers for the biharmonic nonlinear Schrodinger functional $$\begin{aligned} \mathcal {E}_a(u)=\int \limits _{\mathbb {R}^d} |\Delta u(x)|^2 \mathrm{d}x + \int \limits _{\mathbb {R}^d} V(x) |u(x)|^2 \mathrm{d}x - a \int \limits _{\mathbb {R}^d} |u(x)|^{q} \mathrm{d}x \end{aligned}$$ with the mass constraint $$\int |u|^2=1$$ . We focus on the special power $$q=2(1+4/d)$$ , which makes the nonlinear term $$\int |u|^q$$ scales similarly to the biharmonic term $$\int |\Delta u|^2$$ . Our main results are the existence and blowup behavior of the minimizers when a tends to a critical value $$a^*$$ , which is the optimal constant in a Gagliardo–Nirenberg interpolation inequality.

Inspired by Clarkson's inequalities for $L^p$ and continuing work from \cite{CR}, this paper computes the optimal constant $C$ in the weak parallelogram laws $$ \|f + g \|^r + C\|f - g\|^r \leq 2^{r-1}\big( \|f\|^r + \|g\|^r \big), $$ $$ \|f + g \|^r + C\|f- g \|^r \geq 2^{r-1}\big( \|f\|^r + \|g \|^r \big)$$ for the $L^p$ spaces, $1 < p < \infty$., 10 pages

In this paper we obtain the best constants in some higher order Sobolev inequalities in the critical exponent. These inequalities can be separated into two types: those that embed into L ∞ ( R N ) and those that embed into slightly larger target spaces. Concerning the former, we show that for k ∈ { 1 , … , N − 1 } , N − k even, one has an optimal constant c k > 0 such that ‖ u ‖ L ∞ ≤ c k ∫ | ∇ k ( − Δ ) ( N − k ) ∕ 2 u | for all u ∈ C c ∞ ( R N ) (the case k = N was handled in Shafrir, 2018). Meanwhile the most significant of the latter is a variation of D. Adams’ higher order inequality of J. Moser: For Ω ⊂ R N , m ∈ N and p = N m , there exists A > 0 and optimal constant β 0 > 0 such that ∫ Ω exp ( β 0 | u | p ′ ) ≤ A | Ω | for all u such that ‖ ∇ m u ‖ L p ( Ω ) ≤ 1 , where ‖ ∇ m u ‖ L p ( Ω ) is the traditional semi-norm on the space W m , p ( Ω ) .

Let $0 < s < 1$ and $1< p < 2$ be such that $ps < N$ and let $\Omega$ be a bounded domain containing the origin. In this paper we prove the following improved Hardy inequality: Given $1 \le q < p$, there exists a positive constant $C\equiv C(\Omega, q, N, s)$ such that $$ \int\limits_{\mathbb{R}^N}\int\limits_{\mathbb{R}^N} \, \frac{|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}\,dx\,dy - \Lambda_{N,p,s} \int\limits_{\mathbb{R}^N} \frac{|u(x)|^p}{|x|^{ps}}\,dx$$$$\geq C \int\limits_{\Omega}\int\limits_{\Omega}\frac{|u(x)-u(y)|^p}{|x-y|^{N+qs}}dxdy $$ for all $u \in \mathcal{C}_0^\infty({\Omega})$. Here $\Lambda_{N,p,s}$ is the optimal constant in the Hardy inequality （1.1）.

By exploring some geometry of Segre varieties and Veronese varieties, new families of non linear maximum rank distance codes and optimal constant rank codes are provided.

To estimate the optimal constant in Hardy-type inequalities, some variational formulas and approximating procedures are introduced. The known basic estimates are improved considerably. The results are illustrated by typical examples. It is shown that the sharp factor is meaningful for each finite interval and a classical sharp model is re-examined., Comment: 29 pages, 7 figures in Acta Mathematica Sinica, English Series 2015

Based on a recent work of Ding, Wang, and Xiong, we present some cyclotomic constructions of zero-difference balanced (ZDB) functions, which interpret the previous construction proposed by Cai, Zeng, Helleseth, Tang, and Yang. The resulted ZDB functions can be used to generate optimal constant composition codes and perfect difference systems of sets with new parameters.

The mixed principal eigenvalue of p -Laplacian (equivalently, the optimal constant of weighted Hardy inequality in Lp space) is studied in this paper. Several variational formulas for the eigenvalue are presented. As applications of the formulas, a criterion for the positivity of the eigenvalue is obtained. Furthermore, an approximating procedure and some explicit estimates are presented case by case. An example is included to illustrate the power of the results of the paper.

Frame difference families, which can be obtained via a careful use of cyclotomic conditions attached to strong difference families, play an important role in direct constructions for resolvable balanced incomplete block designs. We establish asymptotic existences for several classes of frame difference families. As corollaries new infinite families of 1-rotational $(pq+1,p+1,1)$-RBIBDs over $\mathbb{F}_{p}^+ \times \mathbb{F}_{q}^+$ are derived, and the existence of $(125q+1,6,1)$-RBIBDs is discussed. We construct $(v,8,1)$-RBIBDs for $v\in\{624,1576,2976,5720,5776,10200,14176,24480\}$, whose existence were previously in doubt. As applications, we establish asymptotic existences for an infinite family of optimal constant composition codes and an infinite family of strictly optimal frequency hopping sequences., arXiv admin note: text overlap with arXiv:1702.07500

Rain attenuation prediction models for link analysis are essential for satellite communications. The diurnal variations of the rain attenuation statistics are valuable for the effective system designs. In this work, a simple prediction model for diurnal rain attenuation statistics with its conversion scale is proposed. The method of calculating an approximated optimal constant conversion scale was developed using the acquired rain attenuation statistics for an average year. When such statistics is not available, a reliable rain attenuation model can replace the acquired statistics. Using the 2-year statistics at Nonthaburi, Thailand, the performance of this simple model was analyzed and found to be generally better than the existing model.

Th e eff ects of alternating and constant temperatures and light regimes, osmotic and salt stress and pH were tested on seed germination in four populations of Eryngium caeruleum . Laboratory experiments revealed that the four populations exhibited diff erent responses to alternating temperature and light conditions. Alternating temperature and photoperiod had a greater positive eff ect on seed germination compared to complete darkness. Th e optimal constant temperature within 10 oC to 15 oC for seed germination of each population was determined in a light/dark photoperiod. Seed germination severely decreased under water stress and was completely inhibited at -0.8 MPa osmotic potential. Saline stress sharply decreased germination percentage. Germination was restricted by pH lower and higher than 5 and 8, respectively. Th e information obtained from this study helps to fi ll the gap of knowledge about seed germination requirements of E. caeruleum and enhance our understanding of this species distribution and its potential to develop in stressful and/or new habitats.

An inequality of Hardy and Littlewood for m-homogeneous polynomials on ℓp spaces is valid for p > m: In this note, among other results, we present an optimal version of this inequality for the case p = m and obtain the optimal constant, when restricted to the case of 2-homogeneous polynomials on ℓ2(R2). In an Appendix we justify why, curiously, the optimal exponents of some Hardy-Littlewood type inequalities do not behave smoothly.Mathematics Subject Classication (2010): 47B10, 26D15, 46B25.Key words: Hardy and Littlewood inequalities, multiple summing operators.

We study the large time behavior of solutions v : Ω × ( 0 , ∞ ) → R of the PDE ∂ t ( | v | p − 2 v ) = Δ p v . We show that e ( λ p / ( p − 1 ) ) t v ( x , t ) converges to an extremal of a Poincare inequality on Ω with optimal constant λ p , as t → ∞ . We also prove that the large time values of solutions approximate the extremals of a corresponding “dual” Poincare inequality on Ω. Moreover, our theory allows us to deduce the large time asymptotics of related doubly nonlinear flows involving various boundary conditions and nonlocal operators.

International audience; This paper is devoted to the Moser-Trudinger-Onofri inequality on smooth compact connected Riemannian manifolds. We establish a rigidity result for the Euler-Lagrange equation and deduce an estimate of the optimal constant in the inequality on two-dimensional closed Riemannian manifolds. Compared to existing results, we provide a non-local criterion which is well adapted to variational methods, introduce a nonlinear flow along which the evolution of a functional related with the inequality is monotone and get an integral remainder term which allows us to discuss optimality issues. As an important application of our method, we also consider the non-compact case of the Moser-Trudinger-Onofri inequality on the two-dimensional Euclidean space, with weights. The standard weight is the one that is computed when projecting the two-dimensional sphere using the stereographic projection, but we also give more general results which are of interest, for instance, for the Keller-Segel model in chemotaxis.

The search of the optimal constant for a generalized Wirtinger inequality in an interval consists in minimizing the $p$-norm of the derivative among all functions whose $q$-norm is equal to~1 and whose $(r-1)$-power has zero average. Symmetry properties of minimizers have attracted great attention in mathematical literature in the last decades, leading to a precise characterization of symmetry and asymmetry regions. In this paper we provide a proof of the symmetry result without computer assisted steps, and a proof of the asymmetry result which works as well for local minimizers. As a consequence, we have now a full elementary description of symmetry and asymmetry cases, both for global and for local minima. Proofs rely on appropriate nonlinear variable changes., Comment: 17 pages

In this paper our main results are the multipolar weighted Hardy inequality c ∑ i = 1 n ∫ R N φ 2 | x − a i | 2 d μ ≤ ∫ R N | ∇ φ | 2 d μ + K ∫ R N φ 2 d μ , c ≤ c o , where the functions φ belong to a weighted Sobolev space H μ 1 , and the proof of the optimality of the constant c o = c o ( N ) : = ( N − 2 2 ) 2 . The Gaussian probability measure dμ is the unique invariant measure for Ornstein–Uhlenbeck type operators. This estimate allows us to get necessary and sufficient conditions for the existence of positive solutions to a parabolic problem corresponding to the Kolmogorov operators defined on smooth functions and perturbed by a multipolar inverse square potential L u + V u = ( Δ u + ∇ μ μ ⋅ ∇ u ) + ∑ i = 1 n c | x − a i | 2 u , x ∈ R N , c > 0 , a 1 , … , a n ∈ R N .