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The formation of a positive image of the educational institution provides an increase in the efficiency of its activities and provides an opportunity to meet the needs of stakeholders. The main purpose of the image formation is also to increase competitive advantage, attract investment, establish and expand partnerships. The development of teaching staff is an integral part of forming a positive image of the educational institution, as employees are the main carriers of the brand. The career of educators is a complex multifaceted process, due to the unity of internal (subjective) and external (objective) factors. Internal factors include the living conditions of the future specialist, and external - the peculiarities of career growth (attitude to the profession and awareness of its importance). The desire for self-improvement and self-education are important drivers for the formation of successful careers of employees of educational institutions, ensuring the expansion of their creative potential, cognitive interests, and the formation of a creative personality. Based on the conducted research on building an effective strategy for the development of the employee of the educational institution, each teacher was asked to conduct their SWOT analysis, which allows identifying his/her strengths and weaknesses, as well as external opportunities and threats. The results of the SWOT analysis enable the employee of the educational institution to discover his / her existing potential and to be ready for possible changes in the external environment, which is changeable and fleeting. The strategy of career growth of the teacher is developed, which includes the following stages: conducting SWOT analysis; improvement of professionally important features and qualities; postgraduate education; passing advanced training courses; participation in various pedagogical forums, seminars, conferences, training; research of the advanced pedagogical experience, in particular concerning the use of innovative technologies; constant improvement of scientific and methodical work; forming your portfolio and, as a result, achieving a new level of career growth. Career development leads to a fundamentally new way of life of a teacher - creative self-realization in the profession, which allows identifying their individual and professional capabilities.

In their 2011 paper on the AGT conjecture, Alba, Fateev, Litvinov and Tarnopolsky (AFLT) obtained a closed-form evaluation for a Selberg integral over the product of two Jack polynomials, thereby unifying the well-known Kadell and Hua--Kadell integrals. In this paper we use a variety of symmetric functions and symmetric function techniques to prove generalisations of the AFLT integral. These include (i) an $\mathrm{A}_n$ analogue of the AFLT integral, containing two Jack polynomials in the integrand; (ii) a generalisation of (i) for $\gamma=1$ (the Schur or GUE case), containing a product of $n+1$ Schur functions; (iii) an elliptic generalisation of the AFLT integral in which the role of the Jack polynomials is played by a pair of elliptic interpolation functions; (iv) an AFLT integral for Macdonald polynomials., Comment: 53 pages; v2 contains minor corrections and changes of notation

We employ the barycentric coordinate system to evaluate the inertia tensor of an arbitrary triangular plate of uniform mass distribution. We find that the physical quantities involving the computation are expressed in terms of a single master integral over barycentric coordinates. To expedite the computation in the barycentric coordinates, we employ Lagrange undetermined multipliers. The moment of inertia is expressed in terms of mass, barycentric coordinates of the pivot, and side lengths. The expression is unique and the most compact in comparison with popular expressions that are commonly used in the field of mechanical engineering. A master integral that is necessary to compute the integral over the triangle in the barycentric coordinate system and derivations of the barycentric coordinates of common triangle centers are provided in appendices. We expect that the barycentric coordinates are particularly efficient in computing physical quantities like the electrostatic potential of a triangular charge distribution. We also illustrate a practical experimental design that can be immediately applied to general-physics experiments.

Given a pair of regular quadratic forms over $\mathbb{Q}$ which are in the same genus and a finite set of primes $P$, we show that there is an effective way to determine a rational equivalence between these two quadratic forms which are integral over every prime in $P$. This answers one of the principal questions posed by Conway and Sloane in their book {\em Sphere packings, lattices and groups}, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol 290, Springer-Verlag, New York, 1999; page 402., 12 pages

We present a method to approximate, with controlled and arbitrarily small error, multiple intregrals over the unit cube $$[0,1]^{d}$$ by a single variable integral over [0, 1]. For this, we use the so called $$\delta $$ -uniform curves, which are a particular case of $$\alpha $$ -dense curves. Our main result improves and extends other existing methods on this subject.

In this paper the turnpike phenomenon is studied for problems of optimal control where both pointwise-in-time state and control constraints can appear. We assume that in the objective function, a tracking term appears that is given as an integral over the time-interval $$[0,\, T]$$ [ 0 , T ] and measures the distance to a desired stationary state. In the optimal control problem, both the initial and the desired terminal state are prescribed. We assume that the system is exactly controllable in an abstract sense if the time horizon is long enough. We show that that the corresponding optimal control problems on the time intervals $$[0, \, T]$$ [ 0 , T ] give rise to a turnpike structure in the sense that for natural numbers n if T is sufficiently large, the contribution of the objective function from subintervals of [0, T] of the form $$\begin{aligned} {[}t - t/2^n,\; t + (T-t)/2^n] \end{aligned}$$ [ t - t / 2 n , t + ( T - t ) / 2 n ] is of the order $$1/\min \{t^n, (T-t)^n\}$$ 1 / min { t n , ( T - t ) n } . We also show that a similar result holds for $$\epsilon $$ ϵ -optimal solutions of the optimal control problems if $$\epsilon >0$$ ϵ > 0 is chosen sufficiently small. At the end of the paper we present both systems that are governed by ordinary differential equations and systems governed by partial differential equations where the results can be applied.

We consider an Sp(4,Z) invariant expression involving two factors of the Kawazumi--Zhang (KZ) invariant each of which is a modular graph with one link, and four derivatives on the moduli space of genus two Riemann surfaces. Manipulating it, we show that the integral over moduli space of a linear combination of a modular graph with two links and the square of the KZ invariant reduces to a boundary integral. We also consider an Sp(4,Z) invariant expression involving three factors of the KZ invariant and six derivatives on moduli space, from which we deduce that the integral over moduli space of a modular graph with three links reduces to a boundary integral. In both cases, the boundary term is completely determined by the KZ invariant. We show that both the integrals vanish., Comment: 27 pages, LaTeX; 4 figures

We first prove an identity involving symmetric polynomials. This identity leads us into exploring the geometry of Lagrangian Grassmannians. As an insight applications, we obtain a formula for the integral over the Lagrangian Grassmannian of a characteristic class of the tautological sub-bundle. Moreover, a relation to that over the ordinary Grassmannian and its application to the degree formula for the Lagrangian Grassmannian are given. Finally, we present further applications to the computation of Schubert structure constants and three-point, degree 1, genus 0 Gromov–Witten invariants of the Lagrangian Grassmannian. Some examples together with explicit computations are presented.

Tolerance has been the most important moral characteristic and social value of a person since the inception of human society. Tolerance, which is defined as a spiritual phenomenon, manifests itself in various forms in all areas of life and determines the nature of communication in human behavior. Among them we can show everything, from the most primitive ancient forms of consciousness to forms of artistic and rational thinking. The essence of each of these forms of self-expression is mutual understanding, understanding and empathy. As human knowledge deepened, thoughts and ideas around this problem expanded. The history of the development of tolerance shows that today this moral and legal category is an integral part of the modern way of life.

In this study, a new approach to improve the 3D Cauchy-type integral is presented for faster and more accurate forward modeling of gravity data produced by a sediment-basement interface. The conventional method for calculating the gravity effect of a sedimentary basin is to discretize that into right-rectangular prisms. Its associated volumetric integral over the prisms has computational complexity which makes volumetric integral time-demanding for 3D modeling. A 3D Cauchy-type integral only discretizes the density contrast surface. In fact, it is a surface integral without transcendental functions, which enables fast computation of potential fields. The purpose of the technique is to increase the accuracy of the customary Cauchy-type integral in order to calculate the gravity field over a sedimentary structure which is more likely in real geological structures. To achieve this, the vertical planes located between basement edges and the horizontal reference plane are considered. The accuracy and computational cost is assessed by synthetic gravity data modeling. Three forward functions, namely improved Cauchy-type integral, customary Cauchy-type integral, and volumetric integral, are applied to calculate the gravity field over synthetic sedimentary basins with different geometries. The volumetric integral is set as a benchmark to validate the efficiency of the presented method. Results are analyzed by comparing the dissimilarities of gravity anomalies calculated using the volumetric integral and each of the customary and improved Cauchy-type integrals. The resulting anomaly differences indicate that, compared with the customary Cauchy-type integral, the improved Cauchy-type integral increases the accuracy in calculated gravity anomalies considerably. Furthermore, forward calculations using the improved Cauchy-type integral require approximately the same time as the customary Cauchy-type integral, and are about 50 times faster than the volumetric integral. In addition, the improved Cauchy-type integral gives better results if the edges of the basement are not at an equal level, which is very likely in real geological structures. The new approach is tested on the basement of the Yucca Flat basin to assess the viability of the proposed model in real cases.

We present both necessary and sufficient conditions for a convex closed shape such that for every convex function the average integral over the shape does not exceed the average integral over its boundary. It is proved that this inequality holds for n-dimensional parallelotopes, n-dimensional balls, and convex polytopes having the inscribed sphere (tangent to all its facets) with the centre in the centre of mass of its boundary.

The numerical solutions of boundary integral equations by the Boundary Element Method (BEM) have been applied in several areas of computational engineering and science such as elasticity and fracture mechanics. The BEM formulations often require the evaluation of complex singular and hypersingular integrals. Therefore, BEM requires special integration schemes for singular elements. The Subtraction Singularity Technique (SST) is a general procedure for evaluating such integrals, which allows for the integral over high-order curved boundary elements. The SST regularises these kernels through the Taylor expansion of the integral kernels around the source point. This study presents the expressions from Taylor expansions, which are required for the regularization of singular and hypersingular boundary integral equations of plane linear anisotropic elasticity. These expressions have been implemented into an academic BEM code, which enables the integration over high-order straight and curved boundary elements. The proposed scheme leads to excellent performance. The results obtained by the proposed scheme are in excellent agreement with reference responses available in the literature.

In the recent paper Bürgisser and Lerario (Journal für die reine und angewandte Mathematik (Crelles J), 2016) introduced a geometric framework for a probabilistic study of real Schubert Problems. They denoted by $$\delta _{k,n}$$ δ k , n the average number of projective k-planes in $${\mathbb {R}}\mathrm {P}^n$$ R P n that intersect $$(k+1)(n-k)$$ ( k + 1 ) ( n - k ) many random, independent and uniformly distributed linear projective subspaces of dimension $$n-k-1$$ n - k - 1 . They called $$\delta _{k,n}$$ δ k , n the expected degree of the real Grassmannian $${\mathbb {G}}(k,n)$$ G ( k , n ) and, in the case $$k=1$$ k = 1 , they proved that: $$\begin{aligned} \delta _{1,n}= \frac{8}{3\pi ^{5/2}} \cdot \left( \frac{\pi ^2}{4}\right) ^n \cdot n^{-1/2} \left( 1+{\mathcal {O}}\left( n^{-1}\right) \right) . \end{aligned}$$ δ 1 , n = 8 3 π 5 / 2 · π 2 4 n · n - 1 / 2 1 + O n - 1 . Here we generalize this result and prove that for every fixed integer $$k>0$$ k > 0 and as $$n\rightarrow \infty $$ n → ∞ , we have $$\begin{aligned} \delta _{k,n}=a_k \cdot \left( b_k\right) ^n\cdot n^{-\frac{k(k+1)}{4}}\left( 1+{\mathcal {O}}(n^{-1})\right) \end{aligned}$$ δ k , n = a k · b k n · n - k ( k + 1 ) 4 1 + O ( n - 1 ) where $$a_k$$ a k and $$b_k$$ b k are some (explicit) constants, and $$a_k$$ a k involves an interesting integral over the space of polynomials that have all real roots. For instance: $$\begin{aligned} \delta _{2,n}= \frac{9\sqrt{3}}{2048\sqrt{2\pi }} \cdot 8^n \cdot n^{-3/2} \left( 1+{\mathcal {O}}\left( n^{-1}\right) \right) . \end{aligned}$$ δ 2 , n = 9 3 2048 2 π · 8 n · n - 3 / 2 1 + O n - 1 . Moreover we prove that these numbers belong to the ring of periods intoduced by Kontsevich and Zagier and give an explicit formula for $$\delta _{1,n}$$ δ 1 , n involving a one-dimensional integral of certain combination of Elliptic functions.

For boundary element analysis of the orthotropic potential problems in thin structures, the higher order elements are expected to discretize the boundary. However, the use of the higher order elements leads to more complex forms of the integrands in boundary integral equations. The resulting nearly singular integrals on the higher order elements are difficult to be evaluated when the source point is very close to the integral element. In this paper, a semi-analytic algorithm is presented to evaluate the nearly singular integrals on the quadratic elements in two dimensional (2-D) orthotropic potential problems. By constructing the approximate singular integral kernels, the nearly singular integrals through subtraction technique are transformed into the sum of regular parts and singular parts. Then, the former are calculated by the conventional Gaussian quadrature and the latter are calculated by the analytical integral formulas. Numerical examples demonstrate that the present semi-analytic algorithm is efficient and accurate to calculate the nearly singular integrals on the quadratic elements. Especially, the BEM with the present semi-analytic algorithm is successfully applied to analyzing 2-D orthotropic potential problems in very thin structures.

This chapter systematically reviews the Nystrom method, including its origin, basic principle, implementation, and comparison to the method of moments (MoM). The most distinct feature of the Nystrom method is that it directly replaces the integral operators of integral equations with an algebraic summation under an appropriate quadrature rule when the operators are smooth. This is a process to transform or discretize continuous integral equations into discrete algebraic equations and the corresponding matrix entries can be generated by simply sampling integral kernels at quadrature points without involving numerical integration. In principle, the method is quite simple, i.e. using an algebraic summation to replace an integral over a small regular domain under a quadrature rule so that the continuous integral equations can be discretized into algebraic matrix equations, but the authors emphasize that the singularity treatment or local correction is the main difficulty in the implementation.

The method of moments (MoM) is the primary numerical method for electromagnetic (EM) analysis by integral equation approach. The Nystrom method (NM) has rapidly developed in recent years and could become a strong competitor or alternative to the MoM. The NM is a kind of point-matching method (PMM), but they are actually different. The NM uses a quadrature rule to select matching points within an element while the PMM can randomly choose the matching points. The used quadrature rule allows the NM to transform a continuous integral over a small domain into a discrete summation, leading to a direct evaluation of integrands to generate far-interaction matrix entries. In contrast, the PMM has to rely on numerical integrations to evaluate matrix entries. We propose a hybrid scheme by combining the merits of the two methods and use it to solve EM scattering by open structures, which have an abnormal EM behavior. Numerical examples are presented to illustrate the scheme and its good performance has been verified.

The Cattaneo equations with Caputo–Fabrizio fractional derivative are investigated. A compact finite difference scheme of Crank–Nicolson type is presented and analyzed, which is proved to have temporal accuracy of second order and spatial accuracy of fourth order. Since this derivative is defined with an integral over the whole passed time, conventional direct solvers generally take computational complexity of OMN2 and require memory of OMN, with M and N the number of space steps and time steps, respectively. We develop a fast evaluation procedure for the Caputo–Fabrizio fractional derivative, by which the computational cost is reduced to OMN operations; meanwhile, only OM memory is required. In the end, several numerical experiments are carried out to verify the theoretical results and show the applicability of the fast compact difference procedure.

We consider the $S^3\times S^1$ superconformal index $\mathcal{I}(\tau)$ of 4d $\mathcal{N}=1$ gauge theories. The Hamiltonian index is defined in a standard manner as the Witten index with a chemical potential $\tau$ coupled to a combination of angular momenta on $S^3$ and the $U(1)$ R-charge. We develop the all-order asymptotic expansion of the index as $q = e^{2 \pi i \tau}$ approaches a root of unity, i.e. as $\widetilde \tau \equiv m \tau + n \to 0$, with $m,n$ relatively prime integers. The asymptotic expansion of $\log\mathcal{I}(\tau)$ has terms of the form $\widetilde \tau^k$, $k = -2, -1, 0, 1$. We determine the coefficients of the $k=-2,-1,1$ terms from the gauge theory data, and provide evidence that the $k=0$ term is determined by the Chern-Simons partition function on $S^3/\mathbb{Z}_m$. We explain these findings from the point of view of the 3d theory obtained by reducing the 4d gauge theory on the $S^1$. The supersymmetric functional integral of the 3d theory takes the form of a matrix integral over the dynamical 3d fields, with an effective action given by supersymmetrized Chern-Simons couplings of background and dynamical gauge fields. The singular terms in the $\widetilde \tau \to 0$ expansion (dictating the growth of the 4d index) are governed by the background Chern-Simons couplings. The constant term has a background piece as well as a piece given by the localized functional integral over the dynamical 3d gauge multiplet. The linear term arises from the supersymmetric Casimir energy factor needed to go between the functional integral and the Hamiltonian index., Comment: v3: minor corrections and clarifications added

We determine the correlation energy of BN, SiO$_2$ and ice polymorphs employing a recently developed RPAx (random phase approximation with exchange) approach. The RPAx provides larger and more accurate polarizabilities as compared to the RPA, and captures effects of anisotropy. In turn, the correlation energy, defined as an integral over the density-density response function, gives improved binding energies without the need for error cancellation. Here, we demonstrate that these features are crucial for predicting the relative energies between low- and high-pressure polymorphs of different coordination number as, e.g., between $\alpha$-quartz and stishovite in SiO$_2$, or layered and cubic BN. Furthermore, a reliable (H$_2$O)$_2$ potential energy surface is obtained, necessary for describing the various phases of ice. The RPAx gives results comparable to other high-level methods such as coupled cluster and quantum Monte Carlo, also in cases where the RPA breaks down. Although higher computational cost than RPA we observe a faster convergence with respect to the number of eigenvalues in the response function., Comment: 13 pages + Appendices, 9 figures

Boundary element method incorporated with free-surface Green’s function (FSGF) is commonly used for radiation and diffraction problems of a floating body. In the solution of the boundary integral e...

In this study we use a boundary integral element-based numerical technique to solve the generalized Burger-Fisher equation. The essential feature of this method is the fundamental integral representation of the solution inside the problem domain by means of both the boundary and domain values. The occurrences of domain integrals within the problem arising from nonlinearity as well as the temporal derivative are not avoided or transferred to the boundary. However, unlike the classical boundary element approach, they are resolved within a finite-element-type discrete domain. The utility and correctness of this formulation are proved by comparing the results obtained herein with closed form solutions.

The expansion of a modular graph function on a torus of modulus $\tau$ near the cusp is given by a Laurent polynomial in $y= \pi \Im (\tau)$ with coefficients that are rational multiples of single-valued multiple zeta-values, apart from the leading term whose coefficient is rational and exponentially suppressed terms. We prove that the coefficients of the non-leading terms in the Laurent polynomial of the modular graph function $D_N(\tau)$ associated with a melon graph is free of irreducible multiple zeta-values and can be written as a polynomial in odd zeta-values with rational coefficients for arbitrary $N \geq 0$. The proof proceeds by expressing a generating function for $D_N(\tau)$ in terms of an integral over the Virasoro-Shapiro closed-string tree amplitude.

Let K be a tower of logarithmic extensions over C ( x ). There exists a well-generated tower E of logarithmic extensions containing K. We outline an additive decomposition in E. More precisely, for an element f of E , the additive decomposition computes g , r ∈ E such that f = g' + r with the following two properties: (i) r is minimal in some sense; (ii) f has an integral in E if and only if r = 0. Furthermore, f ∈ K has an elementary integral over K if and only if r is a C -linear combination of some logarithmic derivatives in E. Consequently, we can determine elementary integrability in logarithmic extensions without solving any Risch's equation. This is a generalization of the results in the paper Additive Decompositions in Primitive Extensions by S. Chen, H. Du and Z. Li in Proc. of ISSAC 2018.

A numerically implementable model of a three-dimensional homogeneous random field in a “horizontal” layer 0 < z < H is constructed assuming that the integral of the field with respect to the “vertical” coordinate z has a given infinitely divisible one-dimensional distribution and a given correlation function. An aggregate of n independent elementary horizontal layers of thickness h = H/n shifted vertically by a random variable uniformly distributed in the interval (0, h) is considered as a basic model.

The singularity cancelation is usually of great importance in the calculation of nearly singular integral that is involved in the boundary element analysis of thin structures and in the extended finite element method for crack problems. The well-known distance transformation can be performed for various kinds of singular kernels to compute the near singular integral. However, in the case of that the source point is located nearby the boundary of the integral element, the result is usually very sensitive to the shape of the sub-triangles. A Distance-Sinh combined transformation method based on the generalized Duffy normalization is proposed in this paper to improve the accuracy of the nearly singular integral calculation. Furthermore, a two-level transformation method is proposed to cancel the singularity of the integrand. In first level of the method, the generalized Duffy transformation is implemented. In the transformed integral space (u, v), the singularity of the integrand along u-direction is eliminated by the distance transformation. The singularity of the integrand along the other direction is canceled by the Sinh transformation. Numerical examples are presented to verify the proposed method.

We consider a mathematical model of the treatment of psoriasis on a finite time interval. The model consists of three nonlinear differential equations describing the interrelationships between the concentrations of T-lymphocytes, keratinocytes, and dendritic cells. The model incorporates two bounded timedependent control functions, one describing the suppression of the interaction between T-lymphocytes and keratinocytes and the other the suppression of the interaction between T-lymphocytes and dendritic cells by medication. For this model, we minimize the weighted sum of the total keratinocyte concentration and the total cost of treatment. This weighted sum is expressed as an integral over the sum of the squared controls. Pontryagin’s maximum principle is applied to find the properties of the optimal controls in this problem. The specific controls are determined for various parameter values in the BOCOP-2.0.5 program environment. The numerical results are discussed.

Let A ⊆ B be an extension of commutative rings with identity and T ( A ) (resp., T ( B ) ) be the total quotient ring of A (resp., B). We say that A is a B-Noetherian ring if every ideal I of A with I B = B is finitely generated, and the extension A ⊆ B is called a Noetherian extension if A is B-Noetherian. In this paper, among other things, we introduce the ring A g ( B ) , the global transform of A in B. We then show that if x ∈ A is B-regular (i.e., x B = B ) and R is a ring between A and A g ( B ) , then R / x R is a Noetherian A-module and every ideal of R containing x is finitely generated. We also study some conditions on A and B under which if A is B-Noetherian with dim B A = 1 , then any ring between A and B is B-Noetherian. This is the Krull-Akizuki theorem when A is an integral domain and B = T ( A ) . Finally, we prove that if T ( B ) is integral over T ( A ) and A is integrally closed in B, then every ideal I of A with I B = B is t-invertible, i.e., ( I I − 1 ) t = A .

This paper extends the concept of a normal pair from commutative ring theory to the context of a pair of (associative unital) rings. This is done by using the notion of integrality introduced by Atterton. It is shown that if $$R \subseteq S$$ are rings and $$D=(d_{ij})$$ is an $$n\times n$$ matrix with entries in S, then D is integral (in the sense of Atterton) over the full ring of $$n\times n$$ matrices with entries in R if and only if each $$d_{ij}$$ is integral over R. If $$R \subseteq S$$ are rings with corresponding full rings of $$n\times n$$ matrices $$R_n$$ and $$S_n$$, then $$(R_n,S_n)$$ is a normal pair if and only if (R, S) is a normal pair. Examples are given of a pair $$(\Lambda , \Gamma )$$ of noncommutative (in fact, full matrix) rings such that $$\Lambda \subset \Gamma $$ is (resp., is not) a minimal ring extension; it can be further arranged that $$(\Lambda , \Gamma )$$ is a normal pair or that $$\Lambda \subset \Gamma $$ is an integral extension.

We present sufficient conditions for the existence of a weak solution of the Dirichlet problem for the heat-conduction equation with random action described by an integral over the general stochastic measure.

The family of partial differential equations −eΔu − 2Δ∞u = 0 (e > 0) is studied in a bounded domain Ω for given boundary data. In the case where e = 1, which is closely related to the study of exponentially harmonic maps, we establish existence and uniqueness of a classical solution as the unique minimizer in a closed subset of an Orlicz–Sobolev space of the appropriate energy functional associated to this problem—the integral over Ω of the exponential energy density $$u \mapsto {1 \over 2}\exp \left( {{{\left| {\nabla u} \right|}^2}} \right)$$ . We also explore the connections between the classical solutions of these problems and infinity harmonic and harmonic maps by studying the limiting behavior of the solutions as e → 0+ and e → ∞, respectively. In the former case, we recover a result of Evans and Yu [6].

We deal with an inverse source problem in a time-fractional wave equation. The time-dependent source term is reconstructed using the additional non-invasive measurement in the form of integral over a part of the boundary. We look for the solution and the source term obeying the variational formulation and the equation obtained from applying the measurement on the equation. Using the Rothe method the existence of the solution is proven. The uniqueness is shown in appropriate spaces and some numerical experiments are presented.

Even for holographic theories that obey boundary causality, the full bulk Lorentzian path integral includes metrics that violate this condition. This leads to the following puzzle: The commutator of two field theory operators at spacelike-separated points on the boundary must vanish. However, if these points are causally related in a bulk metric, then the bulk calculation of the commutator will be nonzero. It would appear that the integral over all metrics of this commutator must vanish exactly for holography to hold. This is puzzling since it must also be true if the commutator is multiplied by any other operator. Upon a careful treatment of boundary conditions in holography, we show how the bulk path integral leads to a natural resolution of this puzzle., Comment: 14 pages, 1 figure; v2: matches published version