Rm Min Research Articles

A 14 nm FinFET 128 Mb SRAM With V<formula formulatype="inline"> <tex Notation="TeX">$_{\rm MIN}$</tex></formula> Enhancement Techniques for Low-Power Applications

Two 128 Mb dual-power-supply SRAM chips are fabricated in a 14 nm FinFET technology. A 0.064 <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\mu$</tex> </formula> m <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$^{2}$</tex> </formula> and a 0.080 <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\mu$</tex> </formula> m <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$^{2}$</tex> </formula> 6T SRAM bitcells are designed for high-density (HD) and high-performance (HP) applications. To improve <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\rm V}_{{\rm MIN}}$</tex></formula> of the high-density SRAM, a negative bitline scheme (NBL) is adopted as a write-assist technique. Then, the disturbance-noise reduction (DNR) scheme is proposed as a read-assist circuit to improve the <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\rm V}_{{\rm MIN}}$</tex></formula> of the high-performance SRAM. The 128 Mb 6T-HD SRAM test-chip is fully demonstrated featuring 0.50 <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">${\rm V}_{{\rm MIN}}$</tex></formula> with 200 mV improvement by NBL, and 0.47 <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">${\rm V}_{{\rm MIN}}$</tex></formula> for the 128 Mb 6T-HP with 40 mV improvement by the DNR. Improved <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">${\rm V}_{{\rm MIN}}$</tex></formula> reduces 45.4% and 12.2% power-consumption of the SRAM macro with the help of each assist circuit, respectively.

CANDIDATE CLUSTERS OF GALAXIES ATz&gt; 1.3 IDENTIFIED IN THE SPITZER SOUTH POLE TELESCOPE DEEP FIELD SURVEY

We present 279 galaxy cluster candidates at $z > 1.3$ selected from the 94 deg$^{2}$ Spitzer South Pole Telescope Deep Field (SSDF) survey. We use a simple algorithm to select candidate high-redshift clusters of galaxies based on Spitzer/IRAC mid-infrared data combined with shallow all-sky optical data. We identify distant cluster candidates in SSDF adopting an overdensity threshold that results in a high purity (80%) cluster sample based on tests in the Spitzer Deep, Wide-Field Survey of the Bo\"otes field. Our simple algorithm detects all three $1.4 < z \leq 1.75$ X-ray detected clusters in the Bo\"otes field. The uniqueness of the SSDF survey resides not just in its area, one of the largest contiguous extragalactic fields observed with Spitzer, but also in its deep, multi-wavelength coverage by the South Pole Telescope (SPT), Herschel/SPIRE and XMM-Newton. This rich dataset will allow direct or stacked measurements of Sunyaev-Zel'dovich effect decrements or X-ray masses for many of the SSDF clusters presented here, and enable systematic study of the most distant clusters on an unprecedented scale. We measure the angular correlation function of our sample and find that these candidates show strong clustering. Employing the COSMOS/UltraVista photometric catalog in order to infer the redshift distribution of our cluster selection, we find that these clusters have a comoving number density $n_c = (0.7^{+6.3}_{-0.6}) \times 10^{-7} h^{3} \mathrm{Mpc}^{-3}$ and a spatial clustering correlation scale length $r_0 = (32 \pm 7) h^{-1} \rm{Mpc}$. Assuming our sample is comprised of dark matter halos above a characteristic minimum mass, $M_{{\rm min}}$, we derive that at $z=1.5$ these clusters reside in halos larger than $M_{{\rm min}} = 1.5^{+0.9}_{-0.7} \times 10^{14} h^{-1} M_{\odot}$. (abridged)

ABJ Wilson loops and Seiberg duality

We study supersymmetric Wilson loops in the ${\cal N} = 6$ supersymmetric $U(N_1)_k\times U(N_2)_{-k}$ Chern-Simons-matter (CSM) theory, the ABJ theory, at finite $N_1$, $N_2$ and $k$. This generalizes our previous study on the ABJ partition function. First computing the Wilson loops in the $U(N_1) \times U(N_2)$ lens space matrix model exactly, we perform an analytic continuation, $N_2$ to $-N_2$, to obtain the Wilson loops in the ABJ theory that is given in terms of a formal series and only valid in perturbation theory. Via a Sommerfeld-Watson type transform, we provide a nonperturbative completion that renders the formal series well-defined at all couplings. This is given by ${\rm min}(N_1,N_2)$-dimensional integrals that generalize the "mirror description" of the partition function of the ABJM theory. Using our results, we find the maps between the Wilson loops in the original and Seiberg dual theories and prove the duality. In our approach we can explicitly see how the perturbative and nonperturbative contributions to the Wilson loops are exchanged under the duality. The duality maps are further supported by a heuristic yet very useful argument based on the brane configuration as well as an alternative derivation based on that of Kapustin and Willett.

CONSTRAINTS ON THE MINIMUM ELECTRON LORENTZ FACTOR AND MATTER CONTENT OF JETS FOR A SAMPLE OF BRIGHT FERMI BLAZARS

We fit the (quasi-)simultaneous multi-waveband spectral energy distributions (SEDs) for a sample of low-synchrotron-peaked (LSP) blazars with a one-zone leptonic model. The seed photons that predominantly come from broad line region (BLR) and infrared (IR) molecular torus are considered respectively in external Compton process. We find that the modeling with IR seed photons is systematically better than that with BLR photons based on a $\chi^2$ test, which suggest that $\gamma$-ray emitting region most possibly stay outside the BLR. The minimum electron Lorentz factor, $\gamma_{\rm min}$, is constrained from the modeling for these LSP blazars with good soft X-ray data (ranges from 5 to 160 with a median value of 55), which plays a key role in jet power estimation. Assuming one-to-one ratio of proton and electron, we find that the jet power for LSP blazars is systematically higher than that of FR II radio galaxies at given 151 MHz radio luminosity, $L_{\rm 151MHz}$, even though FR IIs are regarded as same as LSP blazars in unification scheme except the jet viewing angle. The possible reason is that there are some $e^{\pm}$ pairs in the jet of these blazars. If this is the case, we find the number density of $e^{\pm}$ pairs should be several times higher than that of $e^{-}-p$ pairs by assuming the jet power is the same for LSP blazars and FR IIs at given $L_{\rm 151MHz}$.

Number theoretic applications of a class of Cantor series fractal functions. I

Suppose that \({{(P, Q) \in {\mathbb{N}_{2}^\mathbb{N}} \times {\mathbb{N}_{2}^\mathbb{N}}}}\) and x = E 0.E 1 E 2 · · · is the P-Cantor series expansion of \({x \in \mathbb{R}}\). We define $$\psi_{P,Q}(x) := {\sum_{n=1}^{\infty}} \frac{{\rm min}(E_n, q_{n}-1)}{q_1 \cdots q_n}.$$The functions \({\psi_{P,Q}}\) are used to construct many pathological examples of normal numbers. These constructions are used to give the complete containment relation between the sets of Q-normal, Q-ratio normal, and Q-distribution normal numbers and their pairwise intersections for fully divergent Q that are infinite in limit. We analyze the Hölder continuity of \({\psi_{P,Q}}\) restricted to some judiciously chosen fractals. This allows us to compute the Hausdorff dimension of some sets of numbers defined through restrictions on their Cantor series expansions. In particular, the main theorem of a paper by Y. Wang et al. [29] is improved.Properties of the functions \({\psi_{P,Q}}\) are also analyzed. Multifractal analysis is given for a large class of these functions and continuity is fully characterized. We also study the behavior of \({\psi_{P,Q}}\) on both rational and irrational points, monotonicity, and bounded variation. For different classes of ergodic shift invariant Borel probability measures \({\mu_1}\) and \({\mu_2}\) on \({{\mathbb{N}_2^\mathbb{N}}}\), we study which of these properties \({\psi_{P,Q}}\) satisfies for \({\mu_1 \times \mu_2}\)-almost every (P,Q) \({{\in {\mathbb{N}_{2}^{\mathbb{N}}} \times {\mathbb{N}_{2}^{\mathbb{N}}}}}\). Related classes of random fractals are also studied.

Impact of Correlated RF Noise on SiGe HBT Noise Parameters and LNA Design Implications

This paper presents analytical models of SiGe HBT and low-noise amplifier (LNA) noise parameters accounting for high-frequency noise correlation. The models are verified using measurement data and circuit simulation. The impact of noise correlation is shown to be a strong function of base resistance \(r_{\rm b}\) and emitter resistance \(r_{\rm e}\) , which act as both noise sources and resistance. Correlation and roles of \(r_{\rm b}\) and \(r_{\rm e}\) as resistance have opposite effects on minimum noise figure \(N\!F_{\rm min}\) , which explains why a widely used \(N\!F_{\rm min}\) model that neglects correlation and roles of \(r_{\rm b}\) and \(r_{\rm e}\) as resistance sometimes agreed with measurements. The agreement, however, does not hold for noise matching source resistance \(R_{\rm opt}\) , an important parameter for LNAs. With correlation, noise matching condition is better met for impedance matched LNAs.

Stability Constraints Define the Minimum Subthreshold Swing of a Negative Capacitance Field-Effect Transistor

The current-voltage characteristics of a classical field-effect transistor (FET) is dictated by thermal injection of charge carriers over a gate-controlled energy barrier. It is well known that the subthreshold swing <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX">\((S)\) </tex-math></inline-formula> associated with these transistors cannot be reduced below the Boltzmann limit of 60 mV/decade, which in turn defines the lower limit of power dissipation. Therefore, a number of groups have recently proposed to use negative capacitance (NC) gate insulators in FETs to reduce <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX">\(S\) </tex-math></inline-formula> below the Boltzmann limit. The ferroelectric-FET (FE-FET) and suspended-gate FET (SG-FET) are two examples. It is now well accepted that NC-FETs can lower <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX">\(S\) </tex-math></inline-formula> , but given a specific device architecture, the question of whether there is a lower limit of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX">\(S\) </tex-math></inline-formula> (and if so, the conditions that define the limit) has not been answered. In this paper, we: 1) demonstrate that the fundamental constraints of stability and hysteresis-free operation dictate that there is a lower limit of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX">\(S\) </tex-math></inline-formula> associated with each NC-FET technology; 2) provide a general algorithm to calculate the minimum subthreshold swing <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="TeX">\((S_{\rm min})\) </tex-math></inline-formula> ; and 3) illustrate the concept using SG-FET and FE-FET for two different channel configurations, namely, a bulk FET and an FET with constant channel capacitance. Our results broaden the understanding of field-effect action in NC-FETs and should inform research on these devices.

Influence of Rigidity and Knot Complexity on the Knotting of Confined Polymers.

We employ computer simulations and thermodynamic integration to analyze the effects of bending rigidity and slit confinement on the free energy cost of tying knots, ΔFknotting, on polymer chains under tension. A tension-dependent, nonzero optimal stiffness κmin exists, for which ΔFknotting is minimal. For a polymer chain with several stiffness domains, each containing a large amount of monomers, the domain with stiffness κmin will be preferred by the knot. A local analysis of the bending in the interior of the knot reveals that local stretching of chains at the braid region is responsible for the fact that the tension-dependent optimal stiffness has a nonzero value. The reduction in ΔFknotting for a chain with optimal stiffness relative to the flexible chain can be enhanced by tuning the slit width of the 2D confinement and increasing the knot complexity. The optimal stiffness itself is independent of the knot types we considered, while confinement shifts it toward lower values.

UNCOVERING THE INTRINSIC VARIABILITY OF GAMMA-RAY BURSTS

We develop a robust technique to determine the minimum variability timescale for Gamma-ray Burst light curves, utilizing Haar wavelets. Our approach averages over the data for a given GRB, providing an aggregate measure of signal variation while also retaining sensitivity to narrow pulses within complicated time-series. In contrast to previous studies using wavelets, which simply define the minimum timescale in reference to the measurement noise floor, our approach identifies the signature of temporally-smooth features in the wavelet scaleogram and then additionally identifies a break in the scaleogram on longer timescales as signature of a true, temporally-unsmooth light curve feature or features. We apply our technique to the large sample of Swift GRB Gamma-ray light curves and for the first time -- due to the presence of a large number of GRBs with measured redshift -- determine the distribution of minimum variability timescales in the source frame. We find a median minimum timescale for long-duration GRBs in the source frame of $\Delta t_{\rm min}=0.5$ s, with the shortest timescale found being on the order of 10 ms. This short timescale suggests a compact central engine ($3 \times 10^3$ km). We discuss further implications for the GRB fireball model and present a tantalizing correlation between minimum timescale and redshift, which may in part be due to cosmological time-dilation.

The chromatic number of the complementary graph to a simple graph associated with a commutative ring

Let R be a Noetherian ring. We denote by G(R) the simple graph whose vertices are elements of R and in which two distinct vertices x and y are joined by an edge if x − y is a zero-divisor of R. Let \({\overline{G}(R)}\) be the complementary graph to G(R) and \({\overline{\chi}(R)}\) be the chromatic number of the graph \({\overline{G}(R)}\). In this paper, we determine the chromatic number \({\overline{\chi}(R)}\). Let P1, . . . , Pt be all maximal prime divisors of (0). If \({\overline{\chi}(R)}\) is finite, then $$\overline{\chi}(R) = {\rm min} \{|R/P_{i}|; i = 1, 2, \ldots, t\}$$ where \({|R/P_{i}|}\) denotes the number of the set R/Pi.

Characteristic times of biased random walks on complex networks

We consider degree-biased random walkers whose probability to move from a node to one of its neighbors of degree k is proportional to k(α), where α is a tuning parameter. We study both numerically and analytically three types of characteristic times, namely (i) the time the walker needs to come back to the starting node, (ii) the time it takes to visit a given node for the first time, and (iii) the time it takes to visit all the nodes of the network. We consider a large data set of real-world networks and we show that the value of α which minimizes the three characteristic times differs from the value α(min)=-1 analytically found for uncorrelated networks in the mean-field approximation. In addition to this, we found that assortative networks have preferentially a value of α(min) in the range [-1,-0.5], while disassortative networks have α(min) in the range [-0.5,0]. We derive an analytical relation between the degree correlation exponent ν and the optimal bias value α(min), which works well for real-world assortative networks. When only local information is available, degree-biased random walks can guarantee smaller characteristic times than the classical unbiased random walks by means of an appropriate tuning of the motion bias.

How well do third-order aperture mass statistics separate E- and B-modes?

With 3rd-order statistics of gravitational shear it will be possible to extract valuable cosmological information from ongoing and future weak lensing surveys which is not contained in standard 2nd-order statistics, due to the non-Gaussianity of the shear field. Aperture mass statistics are an appropriate choice for 3rd-order statistics due to their simple form and their ability to separate E- and B-modes of the shear. However, it has been demonstrated that 2nd-order aperture mass statistics suffer from E-/B-mode mixing because it is impossible to reliably estimate the shapes of close pairs of galaxies. This finding has triggered developments of several new 2nd-order statistical measures for cosmic shear. Whether the same developments are needed for 3rd-order shear statistics is largely determined by how severe this E-/B-mixing is for 3rd-order statistics. We test 3rd-order aperture mass statistics against E-/B-mode mixing, and find that the level of contamination is well-described by a function of $\theta/\theta_{\rm min}$, with $\theta_{\rm min}$ being the cut-off scale. At angular scales of $\theta > 10 \;\theta_{\rm min}$, the decrease in the E-mode signal due to E-/B-mode mixing is smaller than 1 percent, and the leakage into B-modes is even less. For typical small-scale cut-offs this E-/B-mixing is negligible on scales larger than a few arcminutes. Therefore, 3rd-order aperture mass statistics can safely be used to separate E- and B-modes and infer cosmological information, for ground-based surveys as well as forthcoming space-based surveys such as Euclid.

Degree Sum Condition for k-ordered Hamiltonian Connected Graphs

Let G be a graph on n vertices. If for any ordered set of vertices S = {v 1, v 2, . . . , v k }, that is, the vertices in S appear in order of the sequence v 1, v 2, . . . , v k , there exists a v 1 ? v k (hamiltonian) path containing S in the given order, then G is k-ordered (hamiltonian) connected. Let {u 1, u 2} and {u 3, u 4} be distinct pairs of nonadjacent vertices. When $${G\neq{K_n}}$$ G ? K n and $${G\neq{K_n-e}}$$ G ? K n - e , we define $${\sigma'_4(G)={\rm min}\{d_G(u_1)+d_G(u_2)+d_G(u_3)+d_G(u_4)\}}$$ ? 4 ? ( G ) = min { d G ( u 1 ) + d G ( u 2 ) + d G ( u 3 ) + d G ( u 4 ) } , otherwise set $${\sigma_4'(G)=\infty}$$ ? 4 ? ( G ) = ? . In this paper we will present some sufficient conditions for a graph to be k-ordered connected based on $${\sigma_4'(G)}$$ ? 4 ? ( G ) . As a main result we will show that if $${\sigma_4'(G)\geq{2n+3k-10}}$$ ? 4 ? ( G ) ? 2 n + 3 k - 10 $${(4 \le k \le \frac{n+1}{2})}$$ ( 4 ≤ k ≤ n + 1 2 ) , then G is k-ordered hamiltonian connected. Our outcomes generalize several related results known before.

Verification of Anisotropic s-Wave Superconducting Gap Structure in CeRu2 from Low-Temperature Field-Angle-Resolved Specific Heat Measurements

The field-angle-resolved specific heat C(T,H,phi) of the f-electron superconductor CeRu2 (Tc=6.3 K) has been measured at low temperatures down to 90 mK on two single crystals of slightly different qualities. We reveal that the C(phi) oscillation in a rotating magnetic field, originating from the gap anisotropy, diminishes at low temperatures below the characteristic field H*, as expected for an anisotropic gap without nodes. We also observe the suppression of H* by decreasing the gap anisotropy ratio $\Delta_{\rm min}/\Delta_{\rm max}$, a behavior that has been predicted from a microscopic theory for anisotropic s-wave superconductors. The present technique is established as a powerful tool for investigating minimum-gap structures as well as nodal structures.

Numerical Radii for Tensor Products of Operators

For bounded linear operators A and B on Hilbert spaces H and K, respectively, it is known that the numerical radii of A, B and \({A\otimes B}\) are related by the inequalities \({w(A)w(B)\le w(A\otimes B)\le {\rm min}\{\|A\|w(B), w(A)\|B\|\}}\) . In this paper, we show that (1) if \({w(A\otimes B) = w(A)w(B)}\) , then w(A) = ρ(A) or w(B) = ρ(B), where ρ(·) denotes the spectral radius of an operator, and (2) if A is hyponormal, then \({w(A\otimes B) = w(A)w(B) = \|A\|w(B)}\) . Here (2) confirms a conjecture of Shiu’s and is proven via dilating the hyponormal A to a normal operator N with the spectrum of N contained in that of A. The latter is obtained from the Sz.-Nagy–Foias dilation theory.

On Titchmarsh–Weyl Functions and Eigenfunction Expansions of First-Order Symmetric Systems

We study general (not necessarily Hamiltonian) first-order symmetric system J y′(t)−B(t)y(t) = Δ(t) f(t) on an interval \({\mathcal{I}=[a,b) }\) with the regular endpoint a. It is assumed that the deficiency indices n ±(T min) of the minimal relation T min associated with this system in \({L^2_\Delta(\mathcal{I})}\) satisfy \({n_-(T_{\rm min})\leq n_+(T_{\rm min})}\). We are interested in boundary conditions playing the role similar to that of separated self-adjoint boundary conditions for Hamiltonian systems. Instead we define λ-depending boundary conditions with Nevanlinna type spectral parameter τ = τ(λ) at the singular endpoint b. With this boundary value problem we associate the matrix m-function m(·) of the size \({N_\Sigma = {\rm dim} {\rm ker} (iJ+I)}\). Its role is similar to that of the Titchmarsh–Weyl coefficient for the Hamiltonian system. In turn, it allows one to define the Fourier transform \({V: L^2_\Delta(\mathcal{I}) \to L^2(\Sigma)}\) where Σ (·) is a spectral matrix function of m(·). If V is an isometry, then the (exit space) self-adjoint extension \({\tilde{T}}\) of T min induced by the boundary problem is unitarily equivalent to the multiplication operator in L 2(Σ). Hence the multiplicity of spectrum of \({\tilde{T}}\) does not exceed N Σ. We also parameterize a set of spectral functions Σ(·) by means of the set of boundary parameters τ. Similar parameterizations for various classes of boundary value problems have earlier been obtained by Kac and Krein, Fulton, Hinton and Shaw, and others.

Best proximity point results for MK-proximal contractions on ordered sets

We discuss the existence and uniqueness of solutions to the minimization problem \({{\rm min}_{x \epsilon A} d(x, Tx)}\), where A, B are nonempty subsets of a partially ordered set X endowed with a metric d, and \({T : A \to B}\) is a non-self-mapping satisfying a proximal contraction of Meir–Keeler type (MK-proximal contraction). An iterative algorithm is also provided to approximate the optimal solution. As particular cases of our obtained results, various fixed point theorems on a metric space endowed with a partial order are deduced.

Cosmological constraints on the new holographic dark energy model with action principle

Recently, a New HDE model with action principle was proposed (Li and Miao, arXiv:1210.0966). This model completely solves the causality and circular problems in the original HDE model, and is similar to the original model except a new term that can be interpreted as dark radiation. In this paper, we make further investigations on this model from the aspect of cosmological observations. Numerically, we confirm that the equations of motion force the $L(z=-1)=0$, making the cut-off $aL$ exactly the future event horizon. We also perform detailed analysis on the dynamical properties of the model, divided into the $c<6$ and $c\geq6$ cases ($c$ is a dimensionless parameter which should be decided by the data). From a combination of the present Union2.1+BAO+CMB+$H_0$ data, we find the model yields $\chi^2_{\rm min}=548.798$ (in a non-flat Universe), comparable to the results of the original HDE model (549.461) and the concordant $\Lambda$CDM model (550.354). At 95.4% CL, we get $1.41<c<3.09$ and correspondingly $-2.25<w(z=-1)<-1.39$, implying the Big Rip fate of the Universe at a high confidence level. Besides, for the constraints on dark radiation, we also get a rough estimation $N_{\rm \rm eff}=3.54^{+0.32+0.67}_{\rm -0.45-0.76}$, with the central value slightly larger than the standard value 3.046.

Parabolic Systems with p, q-Growth: A Variational Approach

We consider the evolution problem associated with a convex integrand \({f : \mathbb{R}^{Nn}\to [0,\infty)}\) satisfying a non-standard p, q-growth assumption. To establish the existence of solutions we introduce the concept of variational solutions. In contrast to weak solutions, that is, mappings \({u\colon \Omega_T \to \mathbb{R}^n}\) which solve $$ \partial_tu-{\rm div} Df(Du)=0 $$ weakly in \({\Omega_T}\), variational solutions exist under a much weaker assumption on the gap q − p. Here, we prove the existence of variational solutions provided the integrand f is strictly convex and $$\frac{2n}{n+2} < p \le q < p+1.$$ These variational solutions turn out to be unique under certain mild additional assumptions on the data. Moreover, if the gap satisfies the natural stronger assumption $$ 2\le p \le q < p+ {\rm min}\big \{1,\frac{4}{n} \big \},$$ we show that variational solutions are actually weak solutions. This means that solutions u admit the necessary higher integrability of the spatial derivative Du to satisfy the parabolic system in the weak sense, that is, we prove that $$u\in L^q_{\rm loc}\big(0,T; W^{1,q}_{\rm loc}(\Omega,\mathbb{R}^N)\big).$$

Bond and site percolation in three dimensions

We simulate the bond and site percolation models on a simple-cubic lattice with linear sizes up to L=512, and estimate the percolation thresholds to be p(c)(bond)=0.24881182(10) and p(c)(site)=0.3116077(2). By performing extensive simulations at these estimated critical points, we then estimate the critical exponents 1/ν=1.1410(15), β/ν=0.47705(15), the leading correction exponent y(i)=-1.2(2), and the shortest-path exponent d(min)=1.3756(3). Various universal amplitudes are also obtained, including wrapping probabilities, ratios associated with the cluster-size distribution, and the excess cluster number. We observe that the leading finite-size corrections in certain wrapping probabilities are governed by an exponent ≈-2, rather than y(i)≈-1.2.